EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE
|
|
- Stuart Tucker
- 6 years ago
- Views:
Transcription
1 Version April 30, 2004.Submied o CTU Repors. EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE Per Krysl Universiy of California, San Diego La Jolla, California , USA pkrysl@ucsd.edu Keywords: Time inegraion, rigid body moion, midpoin rule, symplecic Euler, Verle, Newmark, midpoin Lie algorihm We address he design of ime inegraors for mechanical sysems ha are explici in he forcing evaluaions. Our saring poin is he midpoin rule, eiher in he classical form for he vecor space seing, or he Lie form for he roaion group. By inroducing discree, concenraed impulses we can approximae he forcing impressed upon he sysem over he ime sep, and hus arrive a firs-order inegraors. These can be hen composed o yield a second order inegraor wih very desirable properies: sympleciciy and momenum conservaion. 1. Inroducion I is well-known ha he Verle algorihm (explici Newmark for a cerain choice of is parameers) may be wrien as a composiion of wo firs order algorihms, he symplecic Euler and is adjoin [HLG (2003)]. Wha is perhaps less known is ha here is an inerpreaion of his composiion in erms of an approximaion of he midpoin rule, which is of course implici in he evaluaion of he forcing impulse. Our goal in his brief noe is o poin ou ha his mechanically inspired derivaion yields he well-known second order explici Verle algorihm in he vecor space seing, bu also an exremely accurae inegraor for rigid body roaions when he midpoin rule is inerpreed in he Lie sense.
2 2. Vecor space midpoin rule approximaion Le us wrie he iniial value problem for a mechanical sysem wih configuraion u fairly general form as p=f &, p(0) = p 0 u=m & p, u(0) = u0 where p& is he rae of linear momenum, andf = f ( u, ) is he applied force. For simpliciy we shall assume p= Mv, where M is a ime-independen mass marix, and v is he velociy. The midpoin approximaion of he second equaion is u( +) u( ) =M p( +/ 2) where p( + /2) = p( u( + /2), +/ 2) makes his formula implici. To approximae he midpoin momenum, we may use he equaion of moion in inegral form, + h p( + h) = p( ) + f ( τ ) dτ, and numerically evaluae he impulse inegral. This we propose o rea by recourse o he concep of concenraed, discree impulses delivered a pre-seleced ime insans. In paricular, he impulse may be delivered a he end of he ime sep, in which case +/2 f ( τ ) dτ 0. On he oher hand, he impulse may be imposed a he beginning of he ime sep, and +/2 f ( τ) dτ f( ). Therefore, we obain wo algorihms. The firs one, Euler mehod, and he second, *, as is adjoin. n in a, may be recognized as he symplecic p p+ p +f (symplecic Euler) u u+ u +M p+ * p p+ p +f+ (symplecic Euler adjoin) u u+ u +M p These algorihms are symplecic [HLG (2003)], and momenum conserving. Their accuracy is only linear in he ime sep, bu heir composiion preserves boh sympleciciy and momenum conservaion and yields a second order accurae algorihm. Tha algorihm may be recognized as he well-known Verle (explici Newmark wih γ = 1/ 2 ). Ξ = o * /2 /2 (Verle)
3 3. Midpoin rule approximaion on he roaion group Now we shall discuss dynamics of rigid bodies roaing abou a fixed poin. (More deail is available in Reference [K(2004)].) The iniial value problem may be wrien in he conveced descripion (body frame) as Π & = skew[ I ΠΠ ] + T, Π(0) = Π0 R=R & skew[ I Π], R(0) = R0 where Π & is he rae of body frame angular momenum, R is he roaion ensor, T is he applied orque in he body frame, skew[ ] is defined by skew[ h] h = 0, and I is he ime-independen ensor of ineria in he body frame. The second equaion is no in a form suiable for midpoin discreizaion, because he roaion ensor consiues poins of he Lie group SO(3), which is no a vecor space and linear combinaions are no legal operaions on he roaion ensors. To ransform he iniial value problem o a form suiable for our purposes, we shall inroduce he roaion vecor represenaion of he roaion ensor. The equaion of moion is wrien in he spaial frame as π & =RT, where π& is he rae of he spaial angular momenum, and consequenly he equaion of moion may be wrien in inegral form as T Π ( ) = exp[ skew[ Ψ]] Π( 0 ) + R ( ) ( τ ) ( τ) dτ R T 0 where exp[ skew[ Ψ]] = R T ( ) R( 0) is he incremenal roaion hrough vecor Ψ. Upon ime differeniaion and idenificaion wih he original differenial equaion of moion, we obain Ψ& ( ) 1 d exp skew[ ] = Ψ I Π where d exp ( ) is he differenial of he exponenial map. The iniial value problem may skew[ Ψ] be herefore rewrien as Π & = skew[ I ΠΠ ] + T, Π(0) = Π ( skew[ Ψ] ) Ψ& = d exp I Π, Ψ(0) = 0 The midpoin approximaion applied o he second equaion yields ( Ψ = 0 ) Ψ+ = dexp 1 + skew[ ] I Π /2 Ψ 2 which may be simplified by noing d exp 1 Ψ + = Ψ + o give skew[ Ψ+] = +/2 IΨ Π. This equaion needs o be solved for he roaion vecor. Therefore, as for he vecor space seing we ge wo differen algorihms, depending on he chosen approximaion of Π + /2. For he impulse applied a he beginning of he ime sep we obain he SO(3) counerpar of he symplecic Euler inegraor:
4 ( ) Π Π+ exp[ skew( Ψ+ )] Π + T R R+ R exp[skew( Ψ+ )] where Ψ solves IΨ+ = exp[ skew( Ψ+)] ( Π +T) 2 On he oher hand, he oal orque impulse applied a he end of he ime sep yields he adjoin mehod * Π Π+ exp[ skew( Ψ+ )]( Π +T+) R R+ R exp[skew( Ψ+ )] where Ψ solves IΨ+ = exp[ skew( Ψ+)] Π 2 Boh algorihms are firs-order, symplecic, and momenum conserving. As before, he composiion of hese wo algorihms in one ime sep provides us wih a second order accurae algorihm, which is an analogy of he Verle (explici Newmark wih γ = 1/ 2 ) algorihm for he vecor space dynamics Ξ = o * /2 /2 The above algorihms have been called he explici midpoin Lie varian 2 and 1 respecively, and he alernaing midpoin Lie algorihm in Reference [K(2004)]. I bears emphasis ha hese algorihms are no simply he symplecic Euler and is adjoin. They all reduce o he full midpoin Lie algorihm for orque-free moion, and i is only he approximaion of he midpoin momenum evaluaion ha disinguishes hem from he implici midpoin Lie rule. 4. Example The accuracy of he explici midpoin Lie algorihms is raher remarkable as may be seen in Figure 1. We show convergence graphs for he fas spinning heavy op (he kineic energy is dominan, and a numerical mehod has o effecively deal wih precession and nuaion which are moions of disinc frequencies). Even he firs-order mehods perform very well for larger ime seps, and he presen alernaing midpoin Lie algorihm is he sronges performer ou of a selecion of he bes currenly available implici and explici algorihms, including he implici midpoin Lie mehod.
5 Figure 1. Fas Lagrangian op; on he lef hand side convergence in he norm of he error in body-frame angular momenum; on he righ hand side convergence in he norm of he error in he aiude marix: AKW: implici midpoin rule of Ausin, Krishnaprasad, Wang [AKW (1993)]: SW: Simo and Wong [SW (1991)]; NMB: Krysl, Endres Newmark algorihm [KE (2004)]; LIEMID[I]: implici midpoin Lie: LIEMID[E1]: adjoin of he symplecic Euler (explici midpoin Lie varian 1); LIEMID[E2]: symplecic Euler (explici midpoin Lie varian 2); LIEMID[EA]: alernaing explici midpoin Lie mehod. 5. Conclusions We have presened an approach o he consrucion of rigid body inegraors, in paricular for general 3-D roaions, ha are explici in he evaluaion of he forcing, momenum-conserving, and symplecic. The saring poin is he midpoin (implici) rule, which is hen reaed wih numerical inegraion of he forcing wih concenraed impulses. The resuling algorihms conserve momenum, are symplecic, firs-order, and hey are adjoin. Consequenly, heir composiion leads o a second order algorihm, which may be readily inerpreed as a Verle (explici Newmark) inegraor, boh in he vecor space seing, and in he seing of he special orhogonal group of roaions. (We would like o sugges as an appropriae name explici midpoin Lie algorihm for he laer.) For roaional dynamics, his inegraor is a new addiion o he lineup of curren high performance algorihms, and in fac numerical evidence suggess ha i is he bes second-order inegraor o dae. Applicaions abound: molecular dynamics, micro magneics, rigid body dynamics, finie elemen dynamics of deformable solids. Acknowledgemens: Suppor for his research by a Hughes-Chrisensen award is graefully acknowledged. 6. References [AKW (1993)] M. Ausin, P. S. Krishnaprasad, and L. S. Wang. Almos Lie-Poisson inegraors for he rigid body. Journal of Compuaional Physics, 107:105-17, 1993.
6 [HLG (2003)] E. Hairer, C. Lubich, and G. Wanner. Geomeric Numerical Inegraion. Srucure-Preserving Algorihms for Ordinary Differenial Equaions., Springer Series in Compu. Mahemaics, Springer-Verlag, volume 31, [K (2004)] P. Krysl, Explici Momenum-conserving Inegraor for Dynamics of Rigid Bodies Approximaing he Midpoin Lie Algorihm, Inernaional Journal for Numerical Mehods in Engineering, Submied. [KE (2004)] P. Krysl and L. Endres. Explici Newmark/Verle algorihm for ime inegraion of he roaional dynamics of rigid bodies. Inernaional Journal for Numerical Mehods in Engineering, Submied. [SW (1991)] J. C. Simo and K. K. Wong. Uncondiionally sable algorihms for rigid body dynamics ha exacly preserve energy and momenum, Inernaional Journal for Numerical Mehods in Engineering, 31, 19-52, 1991.
Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationFrom Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
More informationUnsteady Flow Problems
School of Mechanical Aerospace and Civil Engineering Unseady Flow Problems T. J. Craf George Begg Building, C41 TPFE MSc CFD-1 Reading: J. Ferziger, M. Peric, Compuaional Mehods for Fluid Dynamics H.K.
More informationThe Contradiction within Equations of Motion with Constant Acceleration
The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationUNCONDITIONALLY STABLE ALGORITHMS FOR RIGID BODY DYNAMICS THAT EXACTLY PRESERVE ENERGY AND MOMENTUM*
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 31, 19-52 (1991) UNCONDITIONALLY STABLE ALGORITHMS FOR RIGID BODY DYNAMICS THAT EXACTLY PRESERVE ENERGY AND MOMENTUM* J. c. SIMO AND K.
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationBasilio Bona ROBOTICA 03CFIOR 1
Indusrial Robos Kinemaics 1 Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationStructural Dynamics and Earthquake Engineering
Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationMulti-scale 2D acoustic full waveform inversion with high frequency impulsive source
Muli-scale D acousic full waveform inversion wih high frequency impulsive source Vladimir N Zubov*, Universiy of Calgary, Calgary AB vzubov@ucalgaryca and Michael P Lamoureux, Universiy of Calgary, Calgary
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationLie Derivatives operator vector field flow push back Lie derivative of
Lie Derivaives The Lie derivaive is a mehod of compuing he direcional derivaive of a vecor field wih respec o anoher vecor field We already know how o make sense of a direcional derivaive of real valued
More informationWe here collect a few numerical tests, in order to put into evidence the potentialities of HBVMs [4, 6, 7].
Chaper Numerical Tess We here collec a few numerical ess, in order o pu ino evidence he poenialiies of HBVMs [4, 6, 7]. Tes problem Le us consider he problem characerized by he polynomial Hamilonian (4.)
More informationModal identification of structures from roving input data by means of maximum likelihood estimation of the state space model
Modal idenificaion of srucures from roving inpu daa by means of maximum likelihood esimaion of he sae space model J. Cara, J. Juan, E. Alarcón Absrac The usual way o perform a forced vibraion es is o fix
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More informationGENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT
Inerna J Mah & Mah Sci Vol 4, No 7 000) 48 49 S0670000970 Hindawi Publishing Corp GENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT RUMEN L MISHKOV Received
More informationThe expectation value of the field operator.
The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More information2001 November 15 Exam III Physics 191
1 November 15 Eam III Physics 191 Physical Consans: Earh s free-fall acceleraion = g = 9.8 m/s 2 Circle he leer of he single bes answer. quesion is worh 1 poin Each 3. Four differen objecs wih masses:
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More information2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
More informationGeneralized Chebyshev polynomials
Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT
More informationLinear Surface Gravity Waves 3., Dispersion, Group Velocity, and Energy Propagation
Chaper 4 Linear Surface Graviy Waves 3., Dispersion, Group Velociy, and Energy Propagaion 4. Descripion In many aspecs of wave evoluion, he concep of group velociy plays a cenral role. Mos people now i
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationOrdinary differential equations. Phys 750 Lecture 7
Ordinary differenial equaions Phys 750 Lecure 7 Ordinary Differenial Equaions Mos physical laws are expressed as differenial equaions These come in hree flavours: iniial-value problems boundary-value problems
More informationAfter the completion of this section the student. Theory of Linear Systems of ODEs. Autonomous Systems. Review Questions and Exercises
Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 V.5 SYSTEMS OF FIRST ORDER LINEAR ODEs Objecives: Afer he compleion of his secion he suden - should recall he definiion of a sysem of linear
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationThe Maxwell Equations, the Lorentz Field and the Electromagnetic Nanofield with Regard to the Question of Relativity
The Maxwell Equaions, he Lorenz Field and he Elecromagneic Nanofield wih Regard o he Quesion of Relaiviy Daniele Sasso * Absrac We discuss he Elecromagneic Theory in some main respecs and specifically
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationA VARIATIONAL APPROACH TO MULTIRATE INTEGRATION FOR CONSTRAINED SYSTEMS
MULTIBODY DYNAMICS, ECCOMAS Themaic Conference J.C. Samin, P. Fisee eds. Brussels, Belgium, 4-7 July A VARIATIONAL APPROACH TO MULTIRATE INTEGRATION FOR CONSTRAINED SYSTEMS Sigrid Leyendecer, Sina Ober-Blöbaum
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationk B 2 Radiofrequency pulses and hardware
1 Exra MR Problems DC Medical Imaging course April, 214 he problems below are harder, more ime-consuming, and inended for hose wih a more mahemaical background. hey are enirely opional, bu hopefully will
More informationLecture Notes 2. The Hilbert Space Approach to Time Series
Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationVariational integrators for constrained dynamical systems
ZAMM Z. Angew. Mah. Mech. 88, No. 9, 677 78 (28) / DOI.2/zamm.2773 Variaional inegraors for consrained dynamical sysems Sigrid Leyendecker,2,, Jerrold E. Marsden,, and Michael Oriz 2, Conrol and Dynamical
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationUCLA Math 135, Winter 2015 Ordinary Differential Equations Review of First-Order Equations from Math 33b
UCLA Mah 35, Winer 05 Ordinary Differenial Equaions Review of Firs-Order Equaions from Mah 33b 7. Numerical Mehods C. David Levermore Deparmen of Mahemaics Universiy of Maryland January 9, 05 7. Numerical
More informationPrinciple of Least Action
The Based on par of Chaper 19, Volume II of The Feynman Lecures on Physics Addison-Wesley, 1964: pages 19-1 hru 19-3 & 19-8 hru 19-9. Edwin F. Taylor July. The Acion Sofware The se of exercises on Acion
More informationContinuous Time Linear Time Invariant (LTI) Systems. Dr. Ali Hussein Muqaibel. Introduction
/9/ Coninuous Time Linear Time Invarian (LTI) Sysems Why LTI? Inroducion Many physical sysems. Easy o solve mahemaically Available informaion abou analysis and design. We can apply superposiion LTI Sysem
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationSingle and Double Pendulum Models
Single and Double Pendulum Models Mah 596 Projec Summary Spring 2016 Jarod Har 1 Overview Differen ypes of pendulums are used o model many phenomena in various disciplines. In paricular, single and double
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationFinite element method for structural dynamic and stability analyses
Finie elemen mehod for srucural dynamic and sabiliy analyses Module- Nonlinear FE Models Lecure-39 Toal and updaed Lagrangian formulaions Prof C Manohar Deparmen of Civil Engineering IIc, Bangalore 56
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationBBP-type formulas, in general bases, for arctangents of real numbers
Noes on Number Theory and Discree Mahemaics Vol. 19, 13, No. 3, 33 54 BBP-ype formulas, in general bases, for arcangens of real numbers Kunle Adegoke 1 and Olawanle Layeni 2 1 Deparmen of Physics, Obafemi
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationRepresenting a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion
More informationConcourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations
Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationKinematics and kinematic functions
Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion
More information0.1 MAXIMUM LIKELIHOOD ESTIMATION EXPLAINED
0.1 MAXIMUM LIKELIHOOD ESTIMATIO EXPLAIED Maximum likelihood esimaion is a bes-fi saisical mehod for he esimaion of he values of he parameers of a sysem, based on a se of observaions of a random variable
More informationShort Introduction to Fractional Calculus
. Shor Inroducion o Fracional Calculus Mauro Bologna Deparameno de Física, Faculad de Ciencias Universidad de Tarapacá, Arica, Chile email: mbologna@ua.cl Absrac In he pas few years fracional calculus
More informationOptimal Path Planning for Flexible Redundant Robot Manipulators
25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp363-368) Opimal Pah Planning for Flexible Redundan Robo Manipulaors H. HOMAEI, M. KESHMIRI Deparmen of Mechanical Engineering
More informationTHE MYSTERY OF STOCHASTIC MECHANICS. Edward Nelson Department of Mathematics Princeton University
THE MYSTERY OF STOCHASTIC MECHANICS Edward Nelson Deparmen of Mahemaics Princeon Universiy 1 Classical Hamilon-Jacobi heory N paricles of various masses on a Euclidean space. Incorporae he masses in he
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationLECTURE 1: GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS
LECTURE : GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS We will work wih a coninuous ime reversible Markov chain X on a finie conneced sae space, wih generaor Lf(x = y q x,yf(y. (Recall ha q
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationSZG Macro 2011 Lecture 3: Dynamic Programming. SZG macro 2011 lecture 3 1
SZG Macro 2011 Lecure 3: Dynamic Programming SZG macro 2011 lecure 3 1 Background Our previous discussion of opimal consumpion over ime and of opimal capial accumulaion sugges sudying he general decision
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationDynamic Analysis of Loads Moving Over Structures
h Inernaional ongress of roaian ociey of echanics epember, 18-, 3 Bizovac, roaia ynamic nalysis of Loads oving Over rucures Ivica Kožar, Ivana Šimac Keywords: moving load, direc acceleraion mehod 1. Inroducion
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationRapid Termination Evaluation for Recursive Subdivision of Bezier Curves
Rapid Terminaion Evaluaion for Recursive Subdivision of Bezier Curves Thomas F. Hain School of Compuer and Informaion Sciences, Universiy of Souh Alabama, Mobile, AL, U.S.A. Absrac Bézier curve flaening
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More information2002 November 14 Exam III Physics 191
November 4 Exam III Physics 9 Physical onsans: Earh s free-fall acceleraion = g = 9.8 m/s ircle he leer of he single bes answer. quesion is worh poin Each 3. Four differen objecs wih masses: m = kg, m
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationChapter 1 Rotational dynamics 1.1 Angular acceleration
Chaper Roaional dynamics. Angular acceleraion Learning objecives: Wha do we mean by angular acceleraion? How can we calculae he angular acceleraion of a roaing objec when i speeds up or slows down? How
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More information