MU+CU+KU=F MU+CU+KU=0
|
|
- Joel Abner Roberts
- 5 years ago
- Views:
Transcription
1 MEEN 67 Handout # MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING ^ Symmetic he motion of a n-dof linea system is descibed by the set of 2 nd ode diffeential equations MU+CU+KU=F t whee U (t) and F (t) ae n ows vectos of displacements and extenal foces, espectively. M, K, C ae the system (nxn) matices of mass, stiffness, and viscous damping coefficients. hese matices ae symmetic, i.e. M=M, K=K, C=C. he solution to Eq. () is detemined uniquely if vectos of initial displacements U o and initial velocities dt ( ) t () V d U o ae specified. Fo fee vibations, the foce vecto F (t) =, and Eq. () is MU+CU+KU= A solution to Eq. (2) is of the fom (2) U e t ψ (3) whee in geneal is a complex numbe. Substitution of Eq. (3) into Eq. (2) leads to the following chaacteistic equation: 2 M C K Ψ f Ψ (4) MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23
2 whee f is a nxn squae matix. he system of homogeneous equations (4) has a nontivial solution only if the deteminant of the system of equation equals zeo, i.e n f c c c2 c3... c2n (5) he oots of the chaacteistic polynomial given by Eq. (5) can be of thee types: a) Real and negative,, coesponding to ove damped modes. b) Puely imaginay, i, fo undamped modes. c) Complex conjugate pais of the fom, id, fo unde damped modes. Clealy if the eal pat of any, it means the system is unstable. he constituent solution, Eq. (3), supeposition of the solution oots satisfying Eq. (4), i.e., U e t ψ can be witten as the e and its associated vectos ψ U 2n t t C Ψ e Ψ Ψ Ψ... Ψ nx2 n n (7) o letting 2 2 (6) wite Eq. (6) as Only if the system is defined by symmetic matices. Othewise, the complex oots may NO BE complex conjugate pais. MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 2
3 U Ψ (8) Ce t t Howeve, a tansfomation of the fom, t 2 nx Unx Ψ q (9) is not possible since this implies the existence of 2n- modal coodinates which is not physically appaent when the numbe of physical coodinates is only n. o ovecome this appaent difficulty, efomulate the poblem in a slightly diffeent fom. Let Y be a 2n- ows vecto composed of the physical velocities and displacements, i.e. U Y= U, and Q= F (t) () be a modified foce vecto. hen wite MU+CU+KU=F t as M U -M U + = M C U K U F o whee (.a) AY+BY=Q M -M A=, B= M C K (.b) (2) MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 3
4 A and B ae 2nx2n matices, symmetic if the M, C, K matices ae also symmetic. Fo fee vibations, Q=, and a solution to Eq. (.b) is sought of the fom: U U =Y=Φ Substitution of Eq. (3) into Eq. (.b) gives: e t (3) A+B Φ (4) which can be witten in the familia fom: DΦ Φ (5) whee - - M M D=-B A= -, o -K M C I D= - - -K M -K C (6) with I as the nxn identity matix. Fom Eq. (5) wite D Ι Φ f Φ (7) he eigenvalue poblem has a nontivial solution if f (8) MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 4
5 Fom Eq. (8) detemine 2n eigenvalues eigenvectos Φ, =, 2.., 2n and associated. Each eigenvecto must satisfy the elationship: DΦ Φ (9) and can be witten as Ψ Φ = 2 Ψ whee Ψ is a nx vecto satisfying: I Ψ Ψ = K M -K CΨ Ψ fom the fist ow of Eq. (2) detemine that: 2 I Ψ Ψ o (2) 2 Ψ Ψ (2) and fom the second ow of Eq. (2), with substitution of the elationship in Eq. (2), obtain -K - MΨ K CΨ Ψ o K M - K C -I Ψ = (23) fo =, 2,.2n. Note that multiplying Eq. (23) by (- K) gives 2 2 M C K Ψ (4) i.e., the oiginal eigenvalue poblem. MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 5
6 Solution of Eq. (9), DΦ Φ, delives the 2n-eigenpais ; Φ Ψ Ψ (24),2,..2n In geneal, the j-components of the eigenvectos numbes witten as Ψ ae complex i a i b e Ψ j j j j j=,2 n j whee and φ denote the magnitude and the phase angle. Note: fo viscous damped systems, not only the amplitudes but also the phase angles ae abitay. Howeve, the atios of amplitudes and diffeences in phase angles ae constant fo each of the elements in the eigenvecto Ψ. hat is, / const and const fo j, k =, 2,..N j k jk j k jk A constituent solution of the homogeneous equation (fee vibation poblem) is then given as: U 2n U t Y= C e Φ (25) Let the (oots) be witten in the fom (when unde-damped) i (26) d and wite Eq. (25) as MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 6
7 2n U Y= U Φ i C e d t (27) and since Ψ Φ Ψ, the vecto of displacements is just U 2n i C Ψ e d t (27) ORHOGONALIY OF DAMPED MODES Each eigenvalue and its coesponding eigenvecto Φ satisfy the equation: ΑΦ B Φ (28) Conside two diffeent eigenvalues (not complex conjugates): s; sand q; q Φ Φ, then if is easy to demonstate that: and infe and Α = Α B=B (a symmetic system), it s q Φ A Φ Φs AΦ q= ; fo s q s q s q Φ ΒΦ = (29) Now, constuct a modal damped matix Φ (2nx2n) fomed by the columns of the modal vectosφ, i.e. Φ Φ Φ... Φ.. Φ Φ (3) 2 n 2n 2n MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 7
8 And wite the othogonality popety as: ΦΑΦ=σ Φ ΒΦ=β (3) Whee σ and β ae (2nx2n) diagonal matices. Now, ecall that the equations of motion in physical coodinates ae : MU+CU+KU=F t ( ) () With the definition U Y U, Eqs. () ae conveted into 2n fist ode diffeential equations: Α Y+Β Y=Q F () t M -M whee A=, B= M C K o uncouple the set of 2n fist-ode Eqs. (32), a solution of the following fom is assumed: 2n Y Φ z ΦZ t t t (32) (2) (33) Substitution of Eq. (33) into Eq. (32) gives: ΑΦΖ+ ΒΦΖ=Q (34) MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 8
9 Pemultiply this equation by the damped modes to get: Φ Α Φ Ζ+ ΦΒΦΖ= Φ Q σζ+β Ζ=G o Φ and use the othogonality popety 2 of Φ Q Eq. (36) epesents a set of 2n uncoupled fist ode equations: z z g t z z g t ( ) ( ) (35) (36) (37) z z g 2N 2N 2N 2N 2N t ( ) Φ ΑΦ Φ B Φ, =, 2..2N whee ; / (38) since ΑΦ Β Φ. In addition, g Φ Q (39) () t () t Initial conditions ae also detemined fom Y U o o Uo with the tansfomation Y Φ Z t σζ =Φ o ΑYo (4.a) 2 he esult below is only valid fo symmetic systems, i.e. with M, K and C as symmetic matices. Fo the moe geneal case (non symmetic system), see the textbook of Meiovitch to find a discussion on LEF and RIGH eigenvectos. MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23 9
10 o o Φ ΑY o =, 2, 2n (4.b) he geneal solution of the fist ode equation z z g t, with initial condition z z t o, is deived fom the Convolution integal with / t t t (4) z z e g e d o Once each of the coodinates to obtain: z () t solutions is obtained, then etun to the physical 2n U Y Φ z ΦZ U t t t (33=43) and since given by: Ψ Φ Ψ, the physical displacement dynamic esponse is U t 2n Ψ z t (44) and the velocity vecto is coespondingly equal to: U t 2n Ψ z t (45) MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23
11 Read/study the accompanying MAHCAD woksheet with a detailed example fo discussion in class. MEEN 67 HD Modal Analysis of MDOF Systems with Viscous Damping L. San Andés 23
12 MODAL ANALYSIS of MDOF linea systems with viscous damping Oiginal by D. Luis San Andes fo MEEN 67 class / SP 8, 2 he equations of motion ae: M d 2 U/dt 2 + C du/dt + K U = F(t) whee M,C,K ae nxn SYMMERIC matices of inetia, viscous damping and stiffness coefficients, and U, du/dt, d 2 U/dt 2,and ae the nx vectos of displacements, velocity and acceleations. F(t) is the nx vecto of genealized foces. Eq () is solved with appopiate initial conditions, at t=, Uo,Vo=dU/dt () ======================================================================== Define elements of inetia, stiffness, and damping matices: m k. 7 c 5 m 2 kg N/m N.s/m k 2. 7 c 2 2 m 3 5 k c 3 n 3 # of DOF Make matices: M m m 2 m 3 K k k 2 k 2 k 2 k 2 k 3 k 3 Initial conditions in displacement and velocity: k 3 k 3 C c c 2 c 2 c 2 c 2 c 3 c 3 c 3 c 3 Uo Vo. PLOs - only N 24# fo time steps natual feqs - undamped damped eigenvals ===================================================================== 2. Evaluate the damped eigenvalues: Rewite Eq (), as A dy/dt + B Y = Q(t), whee Y = [ du/dt, U ], and Q=[, F(t)] ae 2n ow vectos of (velocity, displacements ) and genealized foces; and initial conditions Yo = [ Vo, Uo ] M A = B = M C M K ae 2nx2n Symmetic matices
13 2. define the A & B augmented matices: zeo identity( n) identity( n) the (nxn) null matix A stack( augment( zeo M) augment( M C) ) B stack( augment( M zeo) augment( zeo K) ) 2.2 Use MAHCAD function to calculate eigenvectos and eigenvalues of the genealized eigenvalue poblem, M X = N X. In vibations poblems we set the poblem as: + B =, hence M=-B, N=A to use popely the MAHCAD functions genvecs & genvals genvals( B A) ad/s D genvecs( B A) i i i i i i Recall the undamped natual fequencies note that the (damped) eigenvalues ae complex conjugates, with the same eal pat and +/- imaginay pats D.35.2i.6.2i.64.23i i i i i.6.2i.64.23i i i i i.7.2i.22.44i i i i 4 Note that the eigenvectos ae conjugate pais, i.e. they show the same eal pat and +/- imaginay pat. In addition the fist n-ows of an eigenvecto ae popotional to the 2nd n-ows. he popotionality constant is the damped eingenvalue.
14 2.2. Fom "damped" modal matices using the othogonality popeties: D A D D B D.54.75i.54.75i.26.26i.26.26i.2.i.2.i i i i i i i i i i i i i 4 9.2i i i i i Note off-diagonal tems ae small - but NO zeo (as they should) Check the othogonality popety: compae / atios to eigenvalues jj jj i i i i i i i i i i i i undamped:
15 fo undedamped systems only damping atios: eal pat: imaginay pat: n.5 d = n 2 j 2n j Im j Re j 2.5 nj Re j j dj natual fequencies Im j damped natual feqs. damped eigenvals ( ) damping atios. n d ( natual ) feqs. ad/s damped natual feqs. ( ) ( ) fom undamped modal analysis damped modal esponse
16 3. he 2n fist ode equations A dy/dt + B Y = Q(t) with the tansfomation Y= d Z become 2n equations of the fom: i dz i /dt + i Z i = G i, i=,2,3,...2n whee G= d Q(t) and initial conditions Zo= d A Yo: 3. solve the 2n-fist ode diffeential equations fo the Fee Response to initial conditions, F(t)=: Fom initial conditions in displacement and velocity, set Yo stack( Vo Uo) and in damped modal space: Zo D A Yo p N t p ( p ) t numbe of data points time sequence m 2n get the modal esponse: Z jp Zo j e jt p andback into the physical coodinates: Y= d Z FREE RESPONSE using DAMPED MODES s 2n m Y sp q Ds q Z qp j 2n damped modal esponse
17 Plot the displacements (last n ows of Y vecto): RESPONSE: U ecall: Uo U damped undamped time(secs) RESPONSE: U2.5 U2.5 RESPONSE: U Damped Undamped time(secs) U Damped Undamped time(secs) ( )
18 4. Foced esponse to step load F(t)=Fo and initial conditions Set initial conditions in displacement and velocity: Uo Vo SEP FORCE Fo O Step load esponse hen set: Yo stack( Vo Uo) Qo stack( O Fo) and tansfom to damped modal space: Zo D A Yo Go D Qo max 8 secs f p N and in physical coodinates: SEP RESPONSE using DAMPED MODES t p ( p ) t s m m Y= d Z 2n =================================================================== 6. Fo completeness, obtain also the undamped foced esponse: o Mm M Uo o Mm M Vo Fo t to get the modal esponse max N Z jp j 2n Zo j e jt p Go j e jt p jj m Y sp Ds q q Z qp j n jp o j cos j t p o j sin j t p j j Km jj cos j t p and into physical coodinates: s n SEP RESPONSE using UNDAMPED MODES U ================================================================== Plot the displacements (last n ows of Y vecto): Step load esponse n U sp q sq qp static esponse - check U s K Fo
19 p p RESPONSE: U. U s U damped undamped time(secs) RESPONSE: U2..5 U RESPONSE: U3 Damped Undamped time(secs).5 U Damped Undamped time(secs) ( )
20 5. Peiodic esponse to loading, F(t)=Fo sin(t): Response due to initial conditions vanishes afte a long time because of damping: feq. of excitation Fo fhz 8 2 fhz ecall the undamped fequencies Plot the displacements (last n ows of Y vecto): RESPONSE: U.2 ecall: j U RESPONSE: U2. damped undamped time(secs) U Damped Undamped time(secs) RESPONSE: U3..5 U3.5
21 6) FRF Response to peiodic loading, F=Fo cos(t) UNDAMPED CASE modal foce magntidue: SE and the modal "S-S" esponse is modes. i n Fo i j n max 3 n kmin kmax Fo MODAL esponse k kmin kmax q jk ad/sec <=== focing fequency k k kmax max kmax x cos(t) kmin 2.6 j Km jj k. FRF in physical plane: 2 j 2 s n 2 j k j n U sk i i si q ik x cos(t) hee i=imaginay unit DAMPED CASE set: Qo stack O Fo ( ) focing vecto Go D Qo k kmin kmax k k kmax max ad/s <=== focing fequency j 2n modal esponse Z jk Go j jj k i j x cos(t) and back into the physical plane:
22 s 2n m 2n PERIODIC RESPONSE using DAMPED MODES m Y sk q Ds q Z qk x cos(t) Plot magnitude of esponse in physical space
23 DAMPED Amplitude FRF. Damped physical esponse. Amplitude fequency (ad/s) U U2 U3 UNDAMPED Amplitude FRF UnDamped physical esponse. Amplitude fequency (ad/s) U U2 U3
24 select coodinate to displate physical esponse - damped & undamped jj 2 n 3 DOFs fequency (ad/s) damped undamped LINEAR vetical scale undamped: ( ) fequency (ad/s) damped undamped
MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING
MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens
More informationPHYS 705: Classical Mechanics. Small Oscillations
PHYS 705: Classical Mechanics Small Oscillations Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time
More informationMAC Module 12 Eigenvalues and Eigenvectors
MAC 23 Module 2 Eigenvalues and Eigenvectos Leaning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue poblem by finding the eigenvalues and the coesponding eigenvectos
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationCALCULATING TRANSFER FUNCTIONS FROM NORMAL MODES Revision F
CALCULATING TRANSFER FUNCTIONS FROM NORMAL MODES Revision F By Tom Ivine Email: tom@vibationdata.com Januay, 04 Vaiables F f N H j (f ) i Excitation fequency Natual fequency fo mode Total degees-of-feedom
More informationComplex Eigenvalues. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB omplex Numbes When solving fo the oots of a quadatic equation, eal solutions can not be found when the disciminant is negative. In these
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More informationME 3600 Control Systems Frequency Domain Analysis
ME 3600 Contol Systems Fequency Domain Analysis The fequency esponse of a system is defined as the steady-state esponse of the system to a sinusoidal (hamonic) input. Fo linea systems, the esulting steady-state
More information8 Separation of Variables in Other Coordinate Systems
8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
More informationd 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c
Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.
More informationHydroelastic Analysis of a 1900 TEU Container Ship Using Finite Element and Boundary Element Methods
TEAM 2007, Sept. 10-13, 2007,Yokohama, Japan Hydoelastic Analysis of a 1900 TEU Containe Ship Using Finite Element and Bounday Element Methods Ahmet Egin 1)*, Levent Kaydıhan 2) and Bahadı Uğulu 3) 1)
More informationAPPLICATION OF MAC IN THE FREQUENCY DOMAIN
PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More information1 Spherical multipole moments
Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationPart V: Closed-form solutions to Loop Closure Equations
Pat V: Closed-fom solutions to Loop Closue Equations This section will eview the closed-fom solutions techniques fo loop closue equations. The following thee cases will be consideed. ) Two unknown angles
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More information13. Adiabatic Invariants and Action-Angle Variables Michael Fowler
3 Adiabatic Invaiants and Action-Angle Vaiables Michael Fowle Adiabatic Invaiants Imagine a paticle in one dimension oscillating back and foth in some potential he potential doesn t have to be hamonic,
More informationAppendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More informationPhys101 Lectures 30, 31. Wave Motion
Phys0 Lectues 30, 3 Wave Motion Key points: Types of Waves: Tansvese and Longitudinal Mathematical Repesentation of a Taveling Wave The Pinciple of Supeposition Standing Waves; Resonance Ref: -7,8,9,0,,6,,3,6.
More informationVector d is a linear vector function of vector d when the following relationships hold:
Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationTheWaveandHelmholtzEquations
TheWaveandHelmholtzEquations Ramani Duaiswami The Univesity of Mayland, College Pak Febuay 3, 2006 Abstact CMSC828D notes (adapted fom mateial witten with Nail Gumeov). Wok in pogess 1 Acoustic Waves 1.1
More informationDymore User s Manual Two- and three dimensional dynamic inflow models
Dymoe Use s Manual Two- and thee dimensional dynamic inflow models Contents 1 Two-dimensional finite-state genealized dynamic wake theoy 1 Thee-dimensional finite-state genealized dynamic wake theoy 1
More informationHomework # 3 Solution Key
PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More informationTutorial Exercises: Central Forces
Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total
More informationc n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0.
Poblem {a} Fo t : Ψ(, t ψ(e iet/ h ( whee E mc α (α /7 ψ( e /a πa Hee we have used the gound state wavefunction fo Z. Fo t, Ψ(, t can be witten as a supeposition of Z hydogenic wavefunctions ψ n (: Ψ(,
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationAdvanced Problems of Lateral- Directional Dynamics!
Advanced Poblems of Lateal- Diectional Dynamics! Robet Stengel, Aicaft Flight Dynamics! MAE 331, 216 Leaning Objectives 4 th -ode dynamics! Steady-state esponse to contol! Tansfe functions! Fequency esponse!
More informationChapter 12: Kinematics of a Particle 12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS. u of the polar coordinate system are also shown in
ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Chapte 1 Kinematics of a Paticle A. Bazone 1.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS Pola Coodinates Pola coodinates ae paticlaly sitable fo solving
More informationMath Notes on Kepler s first law 1. r(t) kp(t)
Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is
More information-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.
The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the
More informationQuantum Mechanics II
Quantum Mechanics II Pof. Bois Altshule Apil 25, 2 Lectue 25 We have been dicussing the analytic popeties of the S-matix element. Remembe the adial wave function was u kl () = R kl () e ik iπl/2 S l (k)e
More informationAbsolute Specifications: A typical absolute specification of a lowpass filter is shown in figure 1 where:
FIR FILTER DESIGN The design of an digital filte is caied out in thee steps: ) Specification: Befoe we can design a filte we must have some specifications. These ae detemined by the application. ) Appoximations
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationParticle Systems. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Paticle Systems Univesity of Texas at Austin CS384G - Compute Gaphics Fall 2010 Don Fussell Reading Requied: Witkin, Paticle System Dynamics, SIGGRAPH 97 couse notes on Physically Based Modeling. Witkin
More informationA Relativistic Electron in a Coulomb Potential
A Relativistic Electon in a Coulomb Potential Alfed Whitehead Physics 518, Fall 009 The Poblem Solve the Diac Equation fo an electon in a Coulomb potential. Identify the conseved quantum numbes. Specify
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationExceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs
(May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett@math.umn.edu http://www.math.umn.edu/ gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving
More information= e2. = 2e2. = 3e2. V = Ze2. where Z is the atomic numnber. Thus, we take as the Hamiltonian for a hydrogenic. H = p2 r. (19.4)
Chapte 9 Hydogen Atom I What is H int? That depends on the physical system and the accuacy with which it is descibed. A natual stating point is the fom H int = p + V, (9.) µ which descibes a two-paticle
More informationThe Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07
The Stain Compatibility Equations in Pola Coodinates RAWB Last Update 7//7 In D thee is just one compatibility equation. In D polas it is (Equ.) whee denotes the enineein shea (twice the tensoial shea)
More informationSupplementary material for the paper Platonic Scattering Cancellation for Bending Waves on a Thin Plate. Abstract
Supplementay mateial fo the pape Platonic Scatteing Cancellation fo Bending Waves on a Thin Plate M. Fahat, 1 P.-Y. Chen, 2 H. Bağcı, 1 S. Enoch, 3 S. Guenneau, 3 and A. Alù 2 1 Division of Compute, Electical,
More informationRight-handed screw dislocation in an isotropic solid
Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationClassical Mechanics Homework set 7, due Nov 8th: Solutions
Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with
More informationRelated Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More informationCOMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS
Pogess In Electomagnetics Reseach, PIER 73, 93 105, 2007 COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS T.-X. Song, Y.-H. Liu, and J.-M. Xiong School of Mechanical Engineeing
More informationPhysics 505 Homework No. 9 Solutions S9-1
Physics 505 Homewok No 9 s S9-1 1 As pomised, hee is the tick fo summing the matix elements fo the Stak effect fo the gound state of the hydogen atom Recall, we need to calculate the coection to the gound
More informatione.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6
MOTION IN A PLANE 1. Scala Quantities Physical quantities that have only magnitude and no diection ae called scala quantities o scalas. e.g. Mass, time, speed etc. 2. Vecto Quantities Physical quantities
More informationPerturbation to Symmetries and Adiabatic Invariants of Nonholonomic Dynamical System of Relative Motion
Commun. Theo. Phys. Beijing, China) 43 25) pp. 577 581 c Intenational Academic Publishes Vol. 43, No. 4, Apil 15, 25 Petubation to Symmeties and Adiabatic Invaiants of Nonholonomic Dynamical System of
More informationChapter 2: Introduction to Implicit Equations
Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship
More informationHomework 7 Solutions
Homewok 7 olutions Phys 4 Octobe 3, 208. Let s talk about a space monkey. As the space monkey is oiginally obiting in a cicula obit and is massive, its tajectoy satisfies m mon 2 G m mon + L 2 2m mon 2
More informationChapter 3 Optical Systems with Annular Pupils
Chapte 3 Optical Systems with Annula Pupils 3 INTRODUCTION In this chapte, we discuss the imaging popeties of a system with an annula pupil in a manne simila to those fo a system with a cicula pupil The
More informationNumerical Integration
MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 41 Digital Signal Pocessing Pof. Mak Fowle Note Set #31 Linea Phase FIR Design Optimum Equiipple (Paks-McClellan) Reading: Sect. 1.2.4 1.2.6 of Poakis & Manolakis 1/2 Motivation The window method and
More information3. Electromagnetic Waves II
Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with
More information2 Governing Equations
2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,
More informationyou of a spring. The potential energy for a spring is given by the parabola U( x)
Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium,
More informationNon-Linear Dynamics Homework Solutions Week 2
Non-Linea Dynamics Homewok Solutions Week Chis Small Mach, 7 Please email me at smach9@evegeen.edu with any questions o concens eguading these solutions. Fo the ececises fom section., we sketch all qualitatively
More informationPreliminary Exam: Quantum Physics 1/14/2011, 9:00-3:00
Peliminay Exam: Quantum Physics /4/ 9:-: Answe a total of SIX questions of which at least TWO ae fom section A and at least THREE ae fom section B Fo you answes you can use eithe the blue books o individual
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationEQUATIONS OF MOTION LUCA GUIDO MOLINARI
EQUATIONS OF MOTION LUCA GUIDO MOLINARI 1. Equation of motion of destuction opeatos Conside a system of bosons o femions descibed by a Hamiltonian H = H 1 + H 2, whee H 1 and H 2 ae espectively the one
More informationMA557/MA578/CS557. Lecture 14. Prof. Tim Warburton. Spring
MA557/MA578/CS557 Lectue 4 Sping 3 Pof. Tim Wabuton timwa@math.unm.edu Matlab Notes To ceate a symbolic vaiable (say theta) use the command: theta = sym( theta ); Example manipulation: Matlab cont To evaluate
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationLINEAR PLATE BENDING
LINEAR PLATE BENDING 1 Linea plate bending A plate is a body of which the mateial is located in a small egion aound a suface in the thee-dimensional space. A special suface is the mid-plane. Measued fom
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationINTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More informationUniform Circular Motion
Unifom Cicula Motion Intoduction Ealie we defined acceleation as being the change in velocity with time: a = v t Until now we have only talked about changes in the magnitude of the acceleation: the speeding
More informationProblem Set 10 Solutions
Chemisty 6 D. Jean M. Standad Poblem Set 0 Solutions. Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the lithium atom. You expession should not include any summations (expand them
More informationPHYS 301 HOMEWORK #10 (Optional HW)
PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2
More informationis the instantaneous position vector of any grid point or fluid
Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in
More informationMath Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
More informationsinγ(h y > ) exp(iωt iqx)dωdq
Lectue 9/28/5 Can we ecove a ay pictue fom the above G fo a membane stip? Such a pictue would be complementay to the above expansion in a seies of integals along the many banches of the dispesion elation.
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More information11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.
Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings
More informationA method for solving dynamic problems for cylindrical domains
Tansactions of NAS of Azebaijan, Issue Mechanics, 35 (7), 68-75 (016). Seies of Physical-Technical and Mathematical Sciences. A method fo solving dynamic poblems fo cylindical domains N.B. Rassoulova G.R.
More informationΔt The textbook chooses to say that the average velocity is
1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the
More informationThe Laws of Motion ( ) N SOLUTIONS TO PROBLEMS ! F = ( 6.00) 2 + ( 15.0) 2 N = 16.2 N. Section 4.4. Newton s Second Law The Particle Under a Net Force
SOLUTIONS TO PROBLEMS The Laws of Motion Section 4.3 Mass P4. Since the ca is moving with constant speed and in a staight line, the esultant foce on it must be zeo egadless of whethe it is moving (a) towad
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationf(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4
Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p 2 + 2 (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y
More informationLecture 23. Representation of the Dirac delta function in other coordinate systems
Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.
More information3D-Central Force Problems I
5.73 Lectue #1 1-1 Roadmap 1. define adial momentum 3D-Cental Foce Poblems I Read: C-TDL, pages 643-660 fo next lectue. All -Body, 3-D poblems can be educed to * a -D angula pat that is exactly and univesally
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationMAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS
The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp. 17-1 MGNETIC FIELD
More information