IDENTIFICATION OF MODAL PARAMETERS FROM TRANSMISSIBILITY MEASUREMENTS Patrick Guillaume, Christof Devriendt, and Gert De Sitter Vrije Universiteit Brussel Department of Mechanical Engineering Acoustics & Vibration Research Group avrg@vub.ac.be www.avrg.vub.ac.be Abstract An approach to identify modal parameters from output-only transmissibility measurements is introduced. In general, the poles that are identified from transmissibility measurements do not correspond with the system s poles. However, by combining transmissibility measurements under different loading conditions, it is shown in this paper that model parameters can be identified. 1 Introduction 1.1 Experimental and operational modal analysis During the last decade modal analysis has become a key technology in structural dynamics analysis [1, 2, 3]. Experimental modal analysis (EMA) identifies a modal model, [H(ω)], from the measured forces applied to the test structure, {F (ω)}, and the measured vibration responses {X(ω)}, {X(ω)} =[H(ω)]{F (ω)} (1) with [H(ω)] = N m m=1 {φ m }{L m } T iω λ m + {φ m} {L m } H iω λ m and λ m = σ m + iω dm (3) The modal model (2) expresses the dynamical behavior of the structure as a linear combination of N m resonant modes. Each mode is defined by a damped resonant frequency, f dm = ω dm /2π, a damping ratio, ζ m = σ m / λ m, a mode shape vector, {φ m },andamodal participation vector, {L m }. These modal parameters depend on the geometry, material properties and boundary conditions of the structure. More recently, system identification techniques were developed to identify the modal model from the structure under its operational conditions using output-only data [4, 5, 6, 7]. (2)
These techniques, referred to as operational modal analysis (OMA) or output-only modal analysis, take advantage of the ambient excitation due to e.g. traffic and wind. During an EMA, the structure is often removed from its operating environment and tested in laboratory conditions. The laboratory experimental situation can differ significantly from the real-life operating conditions. An important advantage of OMA is that the structure can remain in its normal operating condition. This allows the identification of more realistic modal models for in-operation structures. Frequency-domain output-only estimators start from power spectra. It can be shown that assuming the operational forces to be white noise sequences the power spectrum matrix or covarinace matrix, [S X (ω)] = cov({x(ω)}), safisfies [S X (ω)] = N m m=1 {φ m }{K m } T iω λ m + {φ m} {K m } H iω λ m {φ m}{k m } T iω + λ m {φ m} {K m } H iω + λ m with {K m } the operational participation vectors, which depend on the modal participation vector, {L m }, and the power spectrum matrix of the unknown operational forces. 1.2 Transmissibilities In this paper attention will be paid to the use of transmissibilities to derive modal parameters. In general, it is not possible to identify modal parameters from transmissibility measurements. Transmissibilities are obtained by taking the ratio of two response spectra, i.e. T ij (ω) = Xi(ω) X j(ω). By assuming a single force that is located in, say, the input degree of freedom (DOF) k, it is readily verified that the transmissibility reduces to T ij (ω) = X i(ω) X j (ω) = H ik(ω)f k (ω) H jk (ω)f k (ω) = N ik(ω) N jk (ω) T ij(ω) k (5) with N ik (ω) andn jk (ω) the numerator polynomials occurring in the transfer-function models H ik = N ik(ω) D(ω) and H jk = N jk(ω) D(ω). Note that the common-denominator polynomial, D(ω), which roots are the system s poles, λ m, disappears by taking the ratio of the two response spectra. Consequently, the poles of the transmissibility function (5) equal the zeroes of transfer function H jk (ω), i.e. the roots of the numerator polynomial N jk (ω). So, in general, the peaks in the magnitude of a transmissibility function do not at all coincide with the resonances of the system. In Section 2 it will be shown that by combining transmissibility measurements under different loading conditions it is possible to identify the modal parameters (i.e., resonant frequencies, damping ratios and mode shape vectors). The procedure will be elaborated and illustrated by means of simulation results in Section 3. Eventually, the conclusions will be drawn in Section 4. 2 Theoretical resuls Making use of the modal model (2) between input DOF, k, and, say, output DOF, i, (4) H ik (ω) = N m m=1 φ im L km iω λ m + φ im L km iω λ m (6)
one concludes that the limit value of the transmissibility function (5) for iω going to the system s poles, λ m,convergesto lim Tij(ω) k = φ iml km = φ im (7) iω λ m φ jm L km φ jm and is independent of the (unknown) force at input DOF k. Consequently, the substraction of two transmissibility functions with the same output DOFs, (i, j), but with different input DOFs, (k, l) satisfies ( lim T k ij (ω) Tij l (ω)) = φ im φ im = (8) iω λ m φ jm φ jm To sum up, the system s poles, λ m, are zeroes of the rational function Tij kl (ω), and, consequently, poles of its inverse, i.e. T l ij Note that the rational function 1 T kl ij k (ω) Tij (ω) 1 Tij kl 1 (ω) Tij kl(ω) = 1 Tij k (ω) T ij l (ω) (9) can be rewritten as 1 T kl ij (ω) = = = 1 H ik (ω) H jk (ω) H il(ω) H jl (ω) H jk (ω)h jl (ω) H ik (ω)h jl (ω) H il (ω)h jk (ω) N jk (ω)n jl (ω) N ik (ω)n jl (ω) N il (ω)n jk (ω) () and so, the order of the polynomial, N ik (ω)n jl (ω) N il (ω)n jk (ω), can exceed the order of the common-denominator polynomial, D(ω). This means that 1 Tij kl (ω) can contain more poles than the system s poles only. Hence, in general, only a subset of the poles of 1 Tij kl (ω) will correspond to the real system s poles. 3 Simulation results Consider the clamped beam given in Figure 1. A single force is applied, one by one, in vertical direction in DOF to 3 and the resulting set of accelerations are computed by means of FEMLAB. Figure 2 shows frequency response functions (FRF) for a force in DOF 1. Figure 1: Clamped beam.
H 11 H 21 H 31 2 3 4 5 6 7 8 9 5 15 2 Figure 2: Frequency response data. Next, the transmissibility between output DOF 2 and 1 is computed and compared in Figure 5 for the 4 different excitation conditions, T21 k,k =,...,3. The vertical dashed lines indicate the location of the resonant peaks of the beam (see Figure 2). One notice that the amplitude s peaks of the transmissibilities do not correspond with the resonant frequencies. However, all transmissibilities cross each other at the resonant frequencies of the beam, which is in agreement with the theoretical results of Section 2. 5 4 3 2 1 2 3 2 3 4 5 5 15 2 Figure 3: Transmissibilities. Starting from transmissibility measurements under different loading conditions, several 1 Tij kl (ω) functions defined in (9) can be computed. These functions are illustrated in Figure 4 for different combinations of the force locations k and l. It can be observed that most of the amplitude s peaks coincide with the resonant peaks of the considered beam. Using a frequency-domain subspace curve fitting approach [7], the frequency response (ω) functions are modeled. The original data are compared to the identified models in Figure 5. The identified damping ratios and damped natural frequencies are summarized in Table 1. Both approaches - frequency response and transmissibility-based - yield identical results. functions as well as the transmissibility-based 1 T kl ij
8 6 4 1 1 1 2 1 3 2 2 4 6 5 15 2 8 6 4 1 12 1 13 1 23 2 2 4 6 5 15 2 Figure 4: Transmissibility-based approach. The transmissibility-based approach results in additional poles that are not related to the system s dynamics. To select the the correct system s poles, a singular value decomposition can, for instance, be used. Consider the following matrix [T (ω)] = T 31(ω) 1 T31(ω) 2 T31(ω) 3 T21 1 (ω) 2 (ω) 3 (ω) (11) 1 1 1 For a weakly-damped system, this matrix should be of rank 1 at the resonant angular frequencies, ω dm, implying that the first 2 singular values σ 1 (ω) andσ 2 (ω) should be close toforω = ω dm.infigure6,1/σ 2 (ω) is plotted as a function of frequency. Only 3 peaks coinciding with the resonant frequencies are present. Once the resonant frequencies are known, it is possible to derive the (operational) mode shapes from the transmissibilities. 4 Conclusions It has been shown in this paper that correct system s poles can be identified starting from transmissibility measurements only. The theoretical results are verified by means of simula-
H 11 H 21 H 31 2 3 4 5 6 7 8 9 5 15 2 (a) FRF-based 8 6 4 1 12 1 13 1 23 2 2 4 6 5 15 2 (b) Transmissibility-based Figure 5: Estimation results. Dotted line: data. Solid line: estimated model. tion data. Classical output-only techniques often require the operational forces to be white noise. This is not necessary for the proposed transmissibility-based approach. The unknown operational forces can be arbitrary (colored noise, swept sine, impact,...) as long as they are persistently exciting in the frequency band of interest. References [1] W. HEYLEN, S. LAMMENS, AND P. SAS, Modal Analysis Theory and Testing. K.U.Leuven, Belgium, 1997. [2] P. VERBOVEN, Frequency domain system identification for modal analysis. PhD Thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel, Belgium, avrg.vub.ac.be, 22.
ζ(h)[%] ζ( 1 T )[%] f d (H) [] f d ( 1 T )[].27.27 8.62 8.62.169.169 53.88 53.88 -.38-98.11.474.474 15.8 15.8 Table 1: Comparison of the estimated damping ratios and damped natural frequencies obtained from the FRF measurements and the transmissibility-based approach. 6 5 1/σ 2 4 3 ampl 2 5 15 2 Figure 6: Selection of the system s poles by means of a sigular value decomposition. [3] B. PEETERS, H. VAN DER AUWERAER, P. GUILLAUME, AND J. LEURIDAN, The PolyMAX frequency-domain method: a new standard for modal parameter estimation? Shock and Vibration, Special Issue dedicated to Professor Bruno Piombo, 11, 39549, 24. [4] L. HERMANS AND H. VAN DER AUWERAER, Modal testing and analysis of structures under operational conditions: industrial applications, Mechanical Systems and Signal Processing, 13(2), 193216, 1999. [5] R. BRINCKER, L. ZHANG, AND P. ANDERSEN, Modal identification of outputonly systems using frequency domain decomposition, Smart Materials and Structures (21) pp. 441445, 21. [6] E. PARLOO, Application of frequency-domain system identification techniques in the field of operational modal analysis. PhD thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel, Belgium, avrg.vub.ac.be, 23. [7] B. CAUBERGHE, Applied frequency-domain system identification in the field of experimental and operational modal analysis. PhD thesis, Department of Mechanical Engineering, Vrije Universiteit Brussel, Belgium, avrg.vub.ac.be, 24.