THE EIGENVALUE PROBLEM

Transcription

1 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 problem is defied as the solutio of the followig equatio: Av = λ v = 1... N (D.1) he N by N A matrix is real ad symmetric; however, it may be sigular ad have zero eigevalues λ. A typical eigevector v has the followig orthogoality properties: v v = 1 ad v v = 0 if m, v Av = λ ad v m Av m = 0 if m therefore (D.2) If all eigevectors V are cosidered, the problem ca be writte as: AV = ΩVΩ or V AV = ΩΩ (D.3) here are may differet umerical methods to solve Equatio (D.3) for eigevectors V ad the diagoal matrix of eigevalues Ω. I structural aalysis, i geeral, it is oly ecessary to solve for the exact eigevalues of small matrices (if subspace iteratio method is used). herefore, the most reliable ad robust will be selected because the computatioal time will always be relatively small. For the determiatio of the dyamic mode shapes ad frequecies of

2 APPENDIX D-2 SAIC AND DYNAMIC ANALYSIS large structural systems, subspace iteratio or Load Depedet Ritz, LDR, vectors are the most efficiet approaches. D.2 HE JACOI MEHOD Oe of the oldest ad most geeral approaches for the solutio of the classical eigevalue problem is the Jacobi method that was first preseted i his is a simple iterative algorithm i which the eigevectors are calculated from the followig series of matrix multiplicatios: ( 0) ( 1) ( 1) ( ) V = (D.4) (0) he startig trasformatio matrix is set to a uit matrix. he iterative (k) orthogoal trasformatio matrix, with four o-zero terms i the i ad j rows ad colums, is of the followig orthogoal form: ii ij = (D.5) ji jj he four o-zero terms are fuctios of a ukow rotatio agle θ ad are defied by: = cosθ ad = siθ (D.6) ii = jj ji = ij herefore, = I, which is idepedet of the agleθ. he typical iteratio ivolves the followig matrix operatio: A ( k 1) = A (D.7)

3 EIGENVALUE PROLEM APPENDIX D-3 he agle is selected to force the terms i,j ad j,i i the matrix his is satisfied if the agle is calculated from: (k) A to be zero. ta 2θ = A 2A ( k 1) ii ( k 1) ij A ( k 1) jj (D.8) he classical Jacobi eigevalue algorithm is summarized withi the computer subroutie give i able D.1. Oe otes that the subroutie for the solutio of the symmetric eigevalue problem by the classical Jacobi method does ot cotai a divisio by ay umber. Also, it ca be proved that after each iteratio cycle, the absolute sum of the off-diagoal terms is always reduced. Hece, the method will always coverge ad yield a accurate solutio for positive, zero or egative eigevalues. he Jacobi algorithm ca be directly applied to all off-diagoal terms, i sequece, util all terms are reduced to a small umber compared to the absolute value of all terms i the matrix. However, the subroutie preseted uses a threshold approach i which it skips the relatively small off-diagoal terms ad operates oly o the large off-diagoal terms. o reduce oe off-diagoal term to zero requires approximately 8N umerical operatios. Clearly, oe caot precisely predict the total umber of umerical operatio because it is a iterative method; however, experiece has idicated that the total umber of umerical operatios to obtai covergece is the order of 10N 3. Assumig a moder (1998) persoal computer ca perform over 6,000,000 operatios per secod, it would require approximately oe secod of computer time to calculate the eigevalues ad eigevectors of a full 100 by 100 matrix.

4 APPENDIX D-4 SAIC AND DYNAMIC ANALYSIS able D.1 Subroutie to Solve the Symmetric Eigevalue Problem SUROUINE JACOI(A,V,NEQ,NF) IMPLICI REAL*8 (A-H,O-Z) DIMENSION A(NEQ,NEQ),V(NEQ,NEQ)!--- EIGENVALUE SOLUION Y JACOI MEHOD ! WRIEN Y ED WILSON July 4, ! A - GIVEN MARIX O E SOLVED ! EIGENVALUES ON DIAGONAL AFER SOLUION! V - MARIX OF EIGENVECORS PRODUCED! NF - NUMER OF SIGNIFICAN FIGURES DAA ZERO /0.0D0/!--- INIALIZE EIGENVECORS ad SE OLERANCE -- OL = 0.1**NF ; SUM = ZERO ; V = ZERO DO I=1,NEQ V(I,I) = 1.0 DO J=1,NEQ ; SUM = SUM + AS(A(I,J)) ; END DO END DO! "I" LOOP IF (SUM.LE.0.0) REURN! NULL MARIX!--- REDUCE MARIX O DIAGONAL SSUM = ZERO DO J=2,NEQ IH = J - 1 DO I=1,IH IF (AS(A(I,J))/SUM.G.OL) HEN! SE A(I,J) O ZERO SSUM = SSUM + AS(A(I,J))! SUM OF OFF-DIAGONAL ERMS!--- CALCULAE ROAION ANGLE AA = AAN2(2.0*A(I,J),A(I,I)-A(J,J))/2.0 SI = SIN(AA) ; CO = COS(AA)!--- MODIFY "I" AND "J" COLUMNS OF "A" AND "V" - DO K=1,NEQ = A(K,I) A(K,I) = CO* + SI*A(K,J) A(K,J) = -SI* + CO*A(K,J) = V(K,I) V(K,I) = CO* + SI*V(K,J) V(K,J) = -SI* + CO*V(K,J) END DO!--- MODIFY DIAGONAL ERMS A(I,I) = CO*A(I,I) + SI*A(J,I) A(J,J) =-SI*A(I,J) + CO*A(J,J) A(I,J) = ZERO!--- MAKE "A" MARIX SYMMERICAL DO K=1,NEQ ; A(I,K)=A(K,I) ; A(J,K)=A(K,J) ; END DO END IF END DO! "J" LOOP END DO! "I" LOOP!--- CHECK FOR CONVERGENCE IF( AS(SSUM)/SUM.G.OL ) GO O 400 REURN ; END

5 EIGENVALUE PROLEM APPENDIX D-5 D.3 CALCULAION OF 3D PRINCIPAL SRESSES he calculatio of the pricipal stresses for a three-dimesioal solid ca be umerically evaluated from the stresses i the x-y-z system by solvig a cubic equatio. However, the defiitio of the directios of the pricipal stresses is ot a simple procedure. A alterative approach to this problem is to write the basic stress trasformatio equatio i terms of the ukow directios of the pricipal stresses i the referece system. Or: σ σ2 0 0 V x1 0 = V x2 σ 3 V x3 V y1 V y2 V y3 V z1 σx V z2 τ yx V z3 τzx τxy σ y τzy τ xz V x1 τ yz V y1 σ z V z1 V x2 V y2 V z2 V x3 V y3 V z3 (D.9) Or, i symbolic form: Ω Ω = V S V (D.10) i which V is the stadard directio cosie matrix. ecause V V is a uit matrix, Equatio (D.3) ca be writte as the followig eigevalue problem: SV = VΩ (D.11) where Ω is a ukow diagoal matrix of the pricipal stresses (eigevalues) ad V is the ukow directio cosie matrix (eigevectors) that uiquely defie the directios of the pricipal stresses. o illustrate the practical applicatio of the classical Jacobi method, cosider the followig state of stress: σ x τxy τxz S = τ yx σ y τ yz = (D.12) τzx τzy σ z he eigevalues, pricipal stresses, ad eigevectors (directio cosies) are:

6 APPENDIX D-6 SAIC AND DYNAMIC ANALYSIS σ σ σ = ad V = (D.13) he solutio of a 3 by 3 eigevalue problem ca be cosidered as a trivial umerical problem. Several hudred of those problems ca be solved by the classical Jacobi method i oe secod of computer time. Note that egative eigevalues are possible. D.4 SOLUION OF HE GENERAL EIGENVALUE PROLEM he geeral eigevalue problem is writte as: AV = Ω VΩ (D.14) where both A ad are symmetrical matrices. he first step is to calculate the eigevectors V of the matrix. We ca ow let the eigevectors V be a liear combiatio or the eigevectors of the matrix. Or: V = ΩV V (D.15) Substitutio of Equatio (D.15) ito Equatio (D.14) ad the pre-multiplicatio of both sides by V yields: V AVV = ΩV V V Ω (D.16) If all eigevalues of the matrix are o-zero, the eigevectors ca be ormalized so that V V = I. Hece, Equatio (D.16) ca be writte i the followig classical form: A V = VΩ (D.17) where A = V AV. herefore, the geeral eigevalue problem ca be solved by applyig the Jacobi algorithm to both matrices. If the matrix is diagoal, the eigevectors V matrix will be diagoal, with the diagoal terms equal to 1 /. his is the case for a lumped mass matrix. Also, mass must be

7 EIGENVALUE PROLEM APPENDIX D-7 associated with all degrees of freedom ad all eigevectors ad values must be calculated. D.5 SUMMARY Oly the Jacobi method has bee preseted i detail i this sectio. raditioally, it has bee restricted to small full matrices i which all eigevalues are required. For the dyamic modal aalysis of very large structural systems, where all the modes are ot required, the basic Jacobi method, coupled with LDR vectors, produces a very robust method. his ew method produces static modes, rigidbody modes ad the classical dyamic modes. hese three types of modes has allowed the Fast Noliear Aalysis, FNA, method to be the fastest ad most accurate solutio approach for a large class of oliear dyamic structural problems. his ew Jacobi method has bee used i SAP2000 for the past te years with impressive results. Carl Gustav Jacob Jacobi (10 December 1804 February 1851) was a Germa mathematicia, who made fudametal cotributios to classical mechaics, dyamics ad astroomy. A crater o the Moo is amed after him. He eared the degree of Doctor of Philosophy i 1825 at erli Uiversity. He was a great teacher ad researcher at the Uiversity of Koigsberg util Oe of his maxims was: 'Ivert, always ivert', expressig his belief that the solutio of may hard problems ca be clarified by re-expressig them i iverse form. He was oe of the early fouders of the theory of determiats; i particular, he iveted the Jacobia determiat formed from the ² differetial coefficiets of give fuctios of idepedet variables, ad which has played a importat part i may aalytical ivestigatios. I 1841 he reitroduced the partial derivative otatio of Legedre, which was to become stadard. hese are mathematical otatios that egieer still use i the developmet of fiite elemets. Jacobi first preseted the method for the determiatio of eigevalues ad eigevectors i 1846.