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sensitivity based method for structural dynamic model improvement

366 R. M. LIN et al.(a) estimated modifications of area S(b) estimated modifications of second moment of area IFig. 20. Estimated modifications of area S and moment of area I (tenth iteration/case 3).5. COMPUTATIONAL CONSIDERATIONSAs shown in the numerical case studies, a completeeigensolution of the updated system is requiredduring each iteration in order to calculate the eigen**sensitivity**and frequency response function **sensitivity**coefficients needed in the case when themeasured coordinates are incomplete. Althoughcomputation is becoming cheaper as more powerfulcomputers are produced at a lower cost, this completeeigensolution is often computationally expensive,especially when systems with big dimensions areconsidered. There**for**e, it is necessary to discusssome computational aspects of the eigenvalue problemso that computational ef**for**t involved can beminimized.If all the eigenvalues and eigenvectors of a matrixare of interest (the complete eigensolution), the LRand QR algorithms [19], which are the most effectiveof known **method**s **for** the genera1 algebraic eigenvalueproblem, can be used. Both **method**s use areduction of the general matrix to triangular **for**m bysimilarity trans**for**ms, but the reduction is achievedby non-unitary trans**for**m in the LR algorithm whileit is achieved by unitary trans**for**m in the QR algorithmwhich is numerically more stable. On the otherhand, if only some of the eigenvalues and theircorresponding eigenvectors are of interest, iterative**method**s can be employed. In fact, in practical finiteelement analysis it is rare **for** all the modes of thesystem to be calculated because, in general, only thelower modes of the system are of interest or evenTable 4. Location of introduced **model**ling errors (case 3)Element No. 26 26 21 1 2 46 47 48 49Between nodes 2425 25-26 26-27 g-l l-2 45-46 46g g-47 4748Modifications (%) 100% (S) 100% (S) 100% (S) -50% (I) -50% (I) -50% (I) -50% (I) - 50% (I) -50% (I)

Structural **dynamic** **model** **improvement** 367valid. The computational cost of solving the eigenvalueproblem is generally proportional to the numberof modes which are required. The most effectivealgorithm used **for** partial eigensolution is the inverseiteration **method** [lS]. In the following we will showhow the inverse iteration **method** can be used effectivelyto reduce the computational ef**for**t involved inthe eigensolution.Let the system to be solved be described by matrix[A] = [M]-‘([K] -t- i[D]) and suppose that only thefirst p modes are of interest. Then, **for** a given matrixtQ,l E C Nxp with orthonormal columns (initial estimateof the required eigenvectors), the followinginverse iteration generates a sequence of matrices{[QJ} E CNxp (k = 1,2, . . . ) which will converge tothe first p eigenvectors of interest**for**k=l,2,...W=f-W’[Q,-,I[QJ[&] = [Z,] (QR factorization of [Z,] tocalculate [Qk] **for** next iteration). (31)After the required eigenvectors [#] (which is theconverged [QJ) are calculated, the ~r~s~ndingeigenvalues [A.] can be found easily **based** on theRayleigh quotient **for**mulationI’A.l= f~lrf~lf~l. (32)The convergence rate of the sequence of (31), asshown in [18], is proportional to the ratio ~,/,l,+, . Itshould be noted that during the iteration process,only one complex inverse is required, that being([k] + i[D])-‘, and in the case of a free-free system inwhich [a is singular, a shift p becomes necessary sothat [A]-’ in (31) becomes ([A] - p[r])-‘. It is easy toprove [20] that system described by [A] and([A - p[Z]) have the same set of eigenvectors andthe corresponding eigenvalues simply differ by avalue of p, Also, in the specific case of the inverseiteration **method** where only one eigenvalue andeigenvector of a system are of interest, p should be sochosen such that it is the closest to the eigenvalue ofinterest [ 181.(a) estimated modifications of area S(b) estimated modifications of second moment of area IFig. 21. Estimated modifications of area S and moment of area I (15th iteration/case 3).

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