The Receptance Approach to Complex Operational Mode Normalization

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1 he Receptance Approach to Complex Operational Mode Normalization Dionisio Bernal Northeastern University, Civil and Environmental Engineering Department, Center for Digital Signal Processing, Boston, MA 19. ABSRAC: Normalization of modes obtained from operational modal analysis has received significant attention since the seminal paper on the use of mass perturbations by Parloo et.al. (). he present paper extends a recently developed formulation known as the Receptance Based Normalization (RBN) scheme to the case of the complex modal model. 1. NRODUCON nput output maps in a modal model are described using eigenvalues, eigenvectors and modal constants. A common notation sets the modal constants to unity and the associated scaled eigenvectors are then said to be normalized. Normalized eigenvectors cannot be obtained from a single test when identification is carried out without deterministic information on the inputs but the information can be obtained by performing complementary tests where the structure is perturbed in a known fashion. ntroduction of the perturbation concept is credited to Parloo et. al.(a), who used estimates of eigenvalue derivatives to extract the information required. he approach in Parloo et al.(a) can run into difficulty when the eigenvalue vs perturbation magnitude is strongly nonlinear because the frequency shift required for accuracy may be too small, given the inherent variability of output-only identification. he obvious modification to deal with this issue is to formulate the problem in terms of total eigenvalue changes and several efforts in this direction have been carried out (Parloo et. al., b, Brinker and Andersen, 3, Bernal 4, 11). his paper extends the normalization scheme recently presented in Bernal (11) to the case where the normalization desired is on the complex modal model.. HE RECEPANCE APPROACH O EGENVECOR NORMALZAON.1 Derivation Let ΔM be the mass perturbation and λ and φ be the pole and arbitrarily scaled (complex) eigenvector in the mass perturbed condition for the th mode quantities without the supra-bar referring to the same quantity in the unperturbed condition. n the derivation that follows it is convenient to treat the eigenvectors as if they are available at all the coordinates, although it will be apparent at the end that only measured coordinates are needed. From the polynomial eigenvalue problem, focusing on the perturbed condition, one has ( ) M + ΔM λ + Cλ + K φ = (1)

2 OMAC'11 4 th nternational Operational Modal Analysis Conference from where it follows that 1 Mλ + Cλ + K ΔM λ φ = φ () nspection of Eq. shows that the inverted matrix is the Receptance of the original system evaluated at s =λ, therefore N φφρ ( l l l ) Δ λφ = φ l= 1 λ λl M (3) where ρ is the square of the constant that is needed to normalize the modes; defining ψ l =φl ρl (4) t follows that the transfer matrix is given by Gs () = N l l= 1 s l ψψ λ l (5) One gathers, therefore, that ψ is the normalized mode and ρ is the square of the normalization constant. For notational convenience we specify this constant as κ, namely κ l = ρl (6) Getting back to eq.3 one notes that with some simple rearranging it can be written as N ( ) l l M l l= 1 λl λ φ φ Δ φ λ ρ =φ (7) Eq.7 describes m equations in the N (squares) of the modal constants formulated using the th polynomial eigenvalue equation of the perturbed system. By combining the equations for each perturbed mode a linear system of the type Qρ = bis obtained. n practice there are only n identified modes so the summation upper limit in Eq.4 has to be taken as n and, after evaluating it at each of the identified modes, the coefficient matrix dimensions is (mxn)xn. Except for approximation resulting from modal truncation the RBN formulation is.. Obtaining the Coefficient Matrices he linear system of equations implicit in Eq.7 can be conveniently formulated as follows: let Φ be the set of identified modes in the original state, at whatever coordinates are measured, and Φ the set in the mass modified state (in both cases including the conugate pairs). he system of equations can be written as Q ρ = vec ( Φ ) (8)

3 3 where {. } ρ = ρ ρ ρ, and 1 n with Φχ1 Φχ Q = (9). Φχn 1 n diag a a a χ = (,,... ) and λ i = ( ) φi Δ φ λi λ a M (1) As can be seen from Eq.8, the constants ρ are coefficients in a proection of the perturbed modes on the basis of the original system. From this perspective one concludes that if the subspace of identified eigenvectors (over all coordinates) has the same span in the original and the perturbed conditions the RBN solution is, independently of truncation. 3. HE SENSVY APPROACH Since we shall use sensitivity to provide contrast for the RBN solution, we derive the formula for the scaling constant of the sensitivity scheme for the case of complex eigenvectors. aking the mass change as Δ M =βm 1, with β a scalar with units of mass and M 1 as a distribution, one has ( ) M +βm1 λ + Cλ+ K ψ= (11) Differentiating Eq.11 with respect to β, pre-multiplying by ψ and evaluating the result at β= one gets, after some simple algebra ( λφ Mφ + φ Cφ) λ + φ M φλ = (1) 1 aking the eigenvector to be the one for which ρ = 1and, using the fact for ρ= 1 one has Balmes(1997) gives λψmψ +ψc ψ = 1 (13) λ+ψ M ψλ = (14) 1 and it follows, adding the subscript for specificity, that ρ = φ λ M1 φ λ On the premise that only two tests are available the derivative of the eigenvalue has to be estimated with the forward difference and one has (15) λ λ ρ φδ φλ M (16)

4 4 OMAC'11 4 th nternational Operational Modal Analysis Conference 4. NUMERCAL EXAMNAONS his section presents numerical results illustrating the performance of RBN as well as some observations relative to the sensitivity solution. he structure is a 16-DOF chain system with a mass distribution [1 1.. etc] and a uniform stiffness equal to 5, in some consistent set of units. he mass change consists of three equal masses located at coordinates {4 8 and 13}, with the magnitude varied to achieve different eigenvalue shifts. Sensors are placed at coordinates {4,8,13 and 16}. he damping matrix is defined by prescribing % modal damping and adding externally grounded dashpots at DOF #1 and # with constants that increase the diagonal entries at these two DOF by a factor of 1. able#1 lists the weighted modal collinearity index (mpcw) for the first 8 modal pairs Pappa et. al.(1993), as well as the poles, the magnitude of the poles, and the damping ratios, defined in standard fashion as the real component of the pole divided by its magnitude. For interest the table also shows the undamped natural frequencies, which are not equal to the magnitude of the poles because the system is not classically damped. As can be seen, the system includes some nearly real modes (1 and ) and some highly complex ones (4-7). able 1. Some parameters of the 16-DOF structure used in the numerical evaluations Mode # (pair) Pole Absolute Value of Pole (rad/sec) Exact undamped frequency (rad/sec) % Damping Ratio Mpcw i i i i i i i i Systematic Bias in the Sensitivity Approach he systematic error in the sensitivity approach derives from the approximation of the derivative with a forward difference. An expression for the percent error in κ due to the nonlinearity of the eigenvalue vs. perturbation is ( λ λ) 1 ϑ= 1 ( ) 1 β λ Plots of ϑ vs. β are feasible but it is more practically meaningful to display the error with respect to the % change in the magnitude of the pole since this is the factor that has to be kept above some limit to prevent variability in the identification to govern the solution. Results showing ϑ as a function of frequency change in percent, for three of the system modes are presented in fig.1. (14)

5 5 ϑ Mode#1 Mode#6 Mode#8 5 R R -5 R % change in λ -1 5 Figure 1: Percent error in the sensitivity estimation of the normalization constant coming from curvature in the eigenvalue vs. perturbation relation. As can be seen, error in the sensitivity estimate in modes 1 and 6 would be small, at reasonable frequency shifts, but unacceptably large in the eighth mode. f high curvatures near the origin in the eigenvalue vs. perturbation are limited to higher modes then the bias in the sensitivity solution is not of much practical significance, given that higher modes are seldom identified. Examination of this matter, however, has not been carried out at the time of writing. 4. Error from Variability in the dentification One anticipates that the bulk of the error in the computation of normalization constants derives from variability in the identification. o provide a pilot examination of this matter in the complex normalization case we look at the relative error in percent in the normalization constant, namely η Rκ ( = ) Rκ ( (k)) η κ ( = ) κ ( (k)) R(k) 1 (k) 1 Rκ ( ) κ ( ) (15a,b) We compute the distribution of η using Monte Carlo simulations with variability simulated as follows: eigenvalues are assigned lognormal distributions with mean values for the real and the imaginary equal to the true values and coefficient of variation of.6, and as.3 respectively. Eigenvector variability is considered by multiplying the magnitude and phase of the eigenvector entries by factors selected from uniform distributions with limits of {.95 and 1.5} and {-.5 and.5} degrees, respectively. We note that when eigenvector error is considered the scaling constant depends on which measured coordinate is used to evaluate it and here, for specificity, we use the location where the (perturbed) mode has the largest amplitude. he mean and the standard deviation of the percent error for the normalization constant of the 8 th mode as a function of frequency shift are depicted in fig.. he top row is the result for RBN and the bottom for sensitivity. n the top row the dotted lines correspond to no truncation and the full lines to computations based on the first 1 eigenvector pairs. he trends are as one anticipates from the theory, namely: in RBN the mean values are small and quite flat, the

6 6 OMAC'11 4 th nternational Operational Modal Analysis Conference downward trend on the real part in the truncated case being the strongest and only reaching about 5% mean error at 5% frequency shift. n the sensitivity the mean error increases sharply with frequency shift, the numerical values consistent with what is anticipates from inspection of fig.1. Since the variability in the eigenvalues becomes less important as the frequency shift increases the standard deviation of the error in both methods decreases as the shift increases. 5 3 (a) 4 (b) 1 mean (η R ) std(η R ) 3 1 mean (η ) std(η ) % freq shift % freq shift (c) mean (η R ) 4 3 (d) mean (η ) 1 std(η R ) 1 std(η ) % freq shift % freq shift Figure : Results for the 8 th mode of a16-dof chain system, top row is RBN and bottom row sensitivity, dotted lines = no truncation, full lines in RBN 1 pairs of complex modes. Based on fig.1 one anticipates that the performance of sensitivity should be adequate on the 6 th mode.his anticipated result is confirmed in fig.3 which shows that the performance of RBN and sensitivity in this case are essentially the same

7 RBN Sensitivity RBN Sensitivity 1 std(η R ) 15 1 std(η ) mean (η R ) 5 mean (η ) % freq shift % freq shift Figure 3: Mean and standard deviations of the percent error defined in eq.15 for the 6 th mode of the 16- DOF system described in the text (RBN based on truncated solution). 5. CONCLUDNG REMARKS he paper presents a generalization of the RBN formulation to the complex modal model and compares results with those obtained using the sensitivity approach. t is noted that since the equation error in the sensitivity approach derives from the difference between the derivative of the eigenvalue (with the respect to the perturbation) and the forward difference estimate, bias depends strongly on the second derivative near the origin. For the variability assigned to eigenvalues and eigenvectors in the numerical example it appears that 3% is a reasonable recommendation for a lower bound on the frequency shift. 6. REFERENCES Balmes E., New results in the identification of normal modes from experimental complex modes, Mechanical Systems and Signal Processing, 11 (), Bernal D., 4. Modal scaling from known mass perturbations, J. Eng. Mech. 13 (9) Bernal D., 11. A Receptance Based Formulation for Modal Scaling Using Mass Perturbations, Mechanical Systems and Signal Processing, 5() Brinker R. and Andersen P., 3. A way of getting scaled mode shapes in output-only modal testing, Proc. of the 1st nt. Modal Anal. Conf., Kissimmee, Fl on CD. Pappa R.S., Elliott K.B., and Schenk A., Consistent-mode indicator for the eigensystem realization algorithm, J. Guidance, Control, and Dynamics, 16 (5), Parloo, E., Verboven, P., Guillaume, P. and Van Overmeire, M., a. Sensitivity-based operational mode shape normalization, Mechanical Systems and Signal Processing, 16, Parloo, E., Verboven, P., Guillaume, P. and Van Overmeire, M., b. terative calculation of nonlinear changes by first-order approximations, Proc. of the th nt. Modal Anal. Conf., Orlando FL.,

IOMAC'15 6 th International Operational Modal Analysis Conference

IOMAC'15 6 th International Operational Modal Analysis Conference 2015 May12-14 Gijón - Spain COMPARISON OF DIFFERENT TECHNIQUES TO SCALE MODE SHAPES IN OPERATIONAL MODAL ANALYSIS. Luis Borja Peral Mtnez