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# sensitivity based method for structural dynamic model improvement

sensitivity based method for structural dynamic model improvement

## 356 R. M. LIN et al.(a)

356 R. M. LIN et al.(a) exact modifications of area 8(b) exact modifications of second moment of area 1Fig. 9. Exact modifications of area S and moment of area I (case 2)2.2. Frequency response function sensitivity In practical measurement, it is often the case thatThe receptance matrix [a(w)] is, by definition,only one column of the receptance matrix isgiven asmeasured. By taking ith column of (16), we have[a(w)] = [-02[M] + [Kll-‘. (14) -=wd,F + F]{a(o)}i.Differentiating (14) with respect to the rth designap, , ,variable p,a[a(w)l-=ap,Upon substitutiona[a(m)l-=ap,-[-d[M] + [K]]-’ -dYIa{a(w)li-=+fg , 1 [-W2[M]+[K]1m'.(15) aprof (14), eqn (15) becomes(17)When structural damping exists, (17) should bemodified asIn the case of the application of frequency functionsensitivity to the practice of analytical model im-(16) provement, which is the topic of the present paper, ithas been found that modification of (18) becomes

necessary in order to take full advantage of thelimited measured data available and to obtain reliableresults. Suppose the analytical mass matrix [MB],stiffness matrix [K,] and the measured receptance{ax(w)>,. are known and that [AM] and [AK] are to beidentified, then the differentiation equation (18) becomesthe corresponding different equation{Aa( = -b,(~)I[--0*Wfl+ PKll{ax(~)Ii,wwhere {Aa( = (aX(w)ji - {aa(w) To illustratethe validity of (19) after multiplication, the righthandside of (19) becomes-ba(~M-~2Wf,1+ [AMI + IKI+WIl) - (I-w*PfJ f Kll>lt~x~~>~iStructural dynamic model improvement 357recommended that the unmeasured coordinates in{a&)>,. are replaced by their analytical counterpartsand then the expanded {a,(w)), be substituted into(22) to calculate the sensitivity coefficients. This willbe discussed in more detail later.3. IMPLEME~ATION OF SENSITIVITY BASED METHODSTO ANALYTICAL MODEL IMPROVEMENT3.1. Eigensensitivity methodFrom the theory of the algebraic eigenvalue problem,a system’s eigenvaiues and eigenvectors areimplicit functions of its design variables. Hence,based on the Taylor series representation, the relationshipbetween the change of modal parameter 65(A< can be the change of any eigenvalue or anyeigenvector element) and the vector of design variablechange {Ap j can be expressed as= -Icc,(~)lfe~j + k@)Ii = CEASE), - ~%@>Ii* At = i s,A/++ i i viiApiApj+..*, (23)(20)i=l ,=1,=1So the ordinary frequency response function sensitivity,calculated based on the analytical model, isa{hi%raP4 a[Kl= -[a,(m)] -02 - + - {a,(w)fi%, 8Pr 1(2fland its modified version isG,(~)Ijap,= -[ff,(w)] -0Jrwhere si and vii are the first- and second-order eigensensitivitycoefficients. The derivation of firstordersensitivity coefficients si for eigenvalues andeigenvectors has been discussed in detail in Sec. 2.1.To be practical, suppose that n (number of measuredcoordinates) out of N (the total number of coordinatesspecified in the analytical model) coordinateshave been measured (n < IV) for the ith mode, thenbased on (23) to a first-order approximation, we haveThe relationship between the ordinary frequencyresponse function ~nsitivity and the modified frequencyresponse function sensitivity is shown schematicallyin Fig. I for a single sensitivity coefficient.When measured coordinates are incomplete, it is-75-100B -125LIzg -fWLtit--*a-300~.IFig. IO. Frequency response functions of analytical and ‘experimental’ models: - analytical, ----‘experimental’.

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