application of frequency-domain system identification techniques in ...

9.3. Applications 251with their unknown operational scaling factor for all considered modes i.φ ref[:,i] = α iψ ref[:,i] (9.35)Although this operation is straightforward for the second order approximation andthe Rayleigh quotient approach, care has to be taken for the multi step first orderapproach. Since this method alters the mode shapes during each step, expression(9.35) should only be substituted in the expression for the first step (n = 0), i.e.,the normalization is effected during the first step. The related expressions thenbecome∂φ (0)[k,i]∂m k∂ω (0)i∂m k≃ −α i ψ [k,i]⎛⎝ α2 i2 ψ2 [k,i] + N m∑ω (1)i = ω (0)= − ωref i2 α2 i ψ[k,i] 2 (9.36)r=1,r≠i∑N ki +k=1(ω ref rφ (1)[k,i] = α iψ [k,i] + ∆φ (0)[k,i] = α iψ [k,i] +(ωi ref ) 2) 2 − (ω refi) 2 α2 r ψ2 [k,r]⎞⎠ (9.37)∂ω (0)i∂m k∆m kN s(9.38)∑N kk=1∂φ (0)[k,i] ∆m k(9.39)∂m k N sFor n > 0 the original expressions (9.15)–(9.18) are retained. For all three methods,the optimization is accomplished by minimizing the following cost functionsfor the second order approximation, the multi step first order method and theRayleigh quotient iteration method respectively.∑N mK 2ND(α 1 , . . . , α Nm ) = ∣ ωmodii=1− (ω modi ) 2ND∣ ∣2(9.40)∑N m∣K MS(α 1 , . . . , α Nm ) =i=1∑N m∣K RQ(α 1 , . . . , α Nm ) =i=1∣ω modi∣ω modi− (ω modi ) MS∣ ∣2− (ω modi ) RQ∣ ∣2(9.41)(9.42)Reasonable starting values for α i can be generated from (4.24) by the original firstorder approach as presented in Chapter 4, Sec. 4.2.2.9.3.2 Sensitivity-based damage identificationAnother related application is sensitivity-based damage identification (Chapter 7).The main idea is to interpret experimentally determined changes in modal parametersin order to obtain information on the structural modifications, i.e., damage,