IOMAC' May Guimarães - Portugal

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1 IOMAC'13 5 th International Operational Modal Analysis Conference 213 May Guimarães - Portugal MODIFICATIONS IN THE CURVE-FITTED ENHANCED FREQUENCY DOMAIN DECOMPOSITION METHOD FOR OMA IN THE PRESENCE OF HARMONIC EXCITATION M.H. Masjedian 1 and M. Keshmiri 2 ABSTRACT Due to the assumption of stochastic input forces, OMA methods normally have limitations and difficulties in the presence of harmonic excitations. Curve-Fitted Enhanced Frequency Domain Decomposition (CFDD) method is a robust OMA method for the system under harmonic excitation. In this method, an estimation of SDOF frequency response function is used to extract modal parameters via curve-fitting in full frequency band. The harmonic components are removed by linear interpolation in SVD graph. Using the entire frequency band to form regression problem causes extra computation and using linear interpolation may cause error in extracted modal parameters especially if a harmonic peak coincides with one of the system natural frequencies. In this paper two modifications are presented to CFDD method. The first modification is using limited data in the vicinity of each mode to form regression problem. The second one is to eliminate the frequency lines corresponding to harmonic components instead of linear interpolation. Using computer simulation of a 4DOF system, accuracy and efficiency of the modified method are compared with the current method. The applicability of the new method is also evaluated using OMA of a steel beam subject to stochastic and harmonic forces. Comparison of the results shows that a satisfactory improvement in the results obtained by the modified method. Keywords: Operational Modal Analysis, Harmonic Excitation, Curve-Fitted Enhanced Frequency Domain Decomposition Method 1. INTRODUCTION In all OMA methods, the inputs are considered to be white-noise, whereas in many applications several harmonic excitations are superposed on the stochastic forces. Most of the OMA methods will fail in the presence of harmonic excitations or will wrongly identified these harmonics as the structural 1 PhD Candidate, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran, m.masjedian@me.iut.ac.ir 2 Associate Professor, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran, mehdik@cc.iut.ac.ir

2 M.H. Masjedian, M. Keshmiri modes. Many researchers adapted OMA methods to consider presence of harmonic excitations. Brincker et al. proposed an indicator for separation of harmonics and structural modes in OMA [1]. The indicator was based on the basic differences of the statistical properties of a harmonic response and a narrow-band stochastic response of a structural mode. Some other methods for separating harmonic excitations and structural modes including short time Fourier transform, singular value decomposition, visual mode shapes comparison, modal assurance criterion, stabilization diagram, and probability density functions are explained in [2]. In 27, Jacobsen et al. applied Enhanced Frequency Domain Decomposition (EFDD) to remove the harmonic components in OMA [3]. Then in 28, Jacobsen and Andersen proposed Curve-fitted Enhanced Frequency Domain Decomposition (CFDD) method to achieve more accurate results compared to the EFDD method [4]. In both methods they used the kurtosis of narrow band-pass filtered signals to identify harmonic components and they removed the harmonic effects by linear interpolation in the singular value graph. In this paper two modifications are presented to the CFDD method to improve OMA in the presence of harmonic excitations. After explanation of the Frequency Domain Decomposition (FDD) method in section 2 and kurtosis indicator in section 3, the MCFDD method is presented in section 4. In section 5 the steps of detection and removing of harmonics is summarized. Then in section 6, accuracy and efficiency of the MCFDD method are compared with the CFDD method using computer simulation of a 4DOF system. In section 7, the effects of the modifications in the CFDD method is investigated with experimental results of a steel beam excited by unknown harmonic and stochastic forces. 2. FREQUENCY DOMAIN DECOMPOSITION METHOD FDD method has been proposed based on Singular Value Decomposition (SVD) of the Power Spectral Density (PSD) of the response signals. In this method modal parameters of a lightly damped structure are obtained using response spectral densities of the system affected by white noise excitations. The relationship between, the PSD matrix of inputs and, the PSD matrix of outputs, can be written as [5]: (1) Singular Value Decomposition of the output PSD is given by: (2) where is the diagonal matrix of the singular values and is the orthogonal matrix of the singular vectors. In FDD method near the k-th peak, the first singular value calculated in the frequency line, is the PSD function of SDOF system corresponding to the k-th mode in the frequency line In this method the peak frequency is considered to be the natural frequency and the first singular vector calculated on this frequency is an estimate of the corresponding mode shape. To estimate more accurate modal parameters in FDD method, the Enhanced FDD (EFDD) method was proposed [6]. In EFDD technique a MAC value is computed for the singular vector corresponding to the peak frequency and the singular vector for each particular frequency line. The values near the k-th peak corresponding to high MAC values are used to construct the k-th SDOF PSD function and the values for other frequencies are set to zero. is taken back to the time domain using the Inverse Discrete Fourier Transform (IDFT). Then, the k-th SDOF correlation function is determined. Natural frequency and damping ratio of this mode are calculated by zero-crossing and logarithmic decrement of. 3. HARMONIC DETECTION BASED ON THE KURTOSIS INDICATOR Kurtosis is defined as the fourth central moment of a stochastic variable normalized with respect to the standard deviation σ as follows[3]: 2

3 5 th International Operational Modal Analysis Conference, Guimarães May 213 where is the mean value of and is denoting the expectation value. If is the response of a structure subject to a stochastic force, its Probability Density Function (PDF) will be normally distributed and its kurtosis becomes 3. But if is the response of the structure subject to a harmonic force, its kurtosis results in 1.5. These values can be used to separate structural and harmonic components[3]. The following steps can be introduced in harmonic detection using kurtosis indicator [3]: For all frequency lines and all measurement channels, the filtered signal is calculated using a narrow band-pass filter around., kurtosis of the filtered signal is computed. the mean of the kurtosis in each frequency line across the measurement channels is calculated. Normally is close to 3 for all frequencies except for harmonic excitation frequencies which is around 1.5. In this method it is necessary to design lots of sharp filters and all the responses should be filtered with these high order filters. Therefore this method is computationally intensive, especially in the case of large number of frequency lines and measurement channels. Andersen et al. [7] introduced an improved method called Fast Kurtosis Checking was proposed using fewer measurement channels and frequency lines. (3) 4. MODIFIED CURVE-FITTED ENHANCED FREQUENCY DOMAIN DECOMPOSITION Jacobsen and Anderson presented the CFDD method as a robust technique to harmonic excitation in OMA [4]. In CFDD method, modal parameters are estimated using curve-fitting in the frequency domain. The main advantage of this method is a more accurate estimation of the natural frequencies and damping ratios especially in the presence of harmonic excitation. In this method, initially the negative lag part of is set to zero, then using DFT, the positive half power spectrum,, is calculated. is an estimation of SDOF FRF corresponding to the k-th mode and it is used to extract modal parameters via curve-fitting in whole frequency band. In this method, for the frequencies outside of the selected range for the k-th mode is set to zero. Therefore, using the entire frequency band to form regression problem causes unneeded computation and error in extracted modal parameters. Also, in EFDD and CFDD methods the harmonic components are removed by linear interpolation in SVD graph. Using linear interpolation may cause error in extracted modal parameters especially if a harmonic peak coincides with a structural natural frequency. To overcome these two shortcomings, two modifications to CFDD method is presented here. In this modified CFDD (MCFDD) method, the regression problem is formed using only selected data for each mode. First using IDFT, is taken back to the time domain and after setting the negative lag part to zero the positive half power spectrum is obtained using DFT. In the vicinity of each peak, is the estimation of SDOF frequency response function of the corresponding mode. The FRF for a SDOF system can be written as [4]: where is the sampling interval. Natural frequency and damping ratio can be extracted from the roots of. Substituting estimated in Eq. (4) results in: (4) 3

4 M.H. Masjedian, M. Keshmiri (5) The regression problem can be formed by rewriting this equation for all selected frequency lines in the vicinity of k-th mode starting from and ending with : (6) To select the frequency lines in the vicinity of each mode, the MAC value is used, as explained in EFDD method in section 2. To ensure that the estimated parameters of to be close to real valued parameters, the regression problem is reformulated as: (7) Finally is calculated using the pseudo-inverse of coefficient matrix: Hence in this modified method the frequency lines corresponding to deterministic components are not participated in formation of regression problem. This is the second modification in CFDD method and it can improve the results comparing with the harmonic removal using linear interpolation. (8) 5. SUMMARY OF THE MCFDD METHOD The steps of proposed method for OMA in the presence of harmonic excitation are summarized as the following: Estimation of the response PSD matrix. Performing singular value decomposition on the PSD matrix in all frequency lines. Calculating the kurtosis index for all frequency lines. Comparing the kurtosis index with a reference number and identifying the harmonic component frequencies. Eliminating the frequency lines corresponding to harmonic component from the. Selecting a frequency range for each mode using MAC value. Using remaining data in selected frequency range to form regression problem in MCFDD method. 4

5 db 5 th International Operational Modal Analysis Conference, Guimarães May 213 Solving the regression problem and extracting the modal parameters for each mode. 6. SIMULATION STUDY In this section the simulation results of a system with known modal parameters are presented. The response of the system subject to stochastic and deterministic inputs is used to assess differences of the CFDD and MCFDD methods. In this assessment the extracted modal parameters are compared with the exact values. A 4DOF mass-spring system with proportional viscous damping is selected for this simulation study. The response of this system subject to stochastic and harmonic forces is calculated for 2 second and is sampled with 1 samples per second rate. The harmonic input frequencies are 3.1 and 5.5 Hz. The SVD plot for this simulation is presented in figure 1. After elimination of the harmonic components using CFDD and MCFDD methods the selected data for the first and second modes are presented along with the fitted curves in figures 2 and 3, respectively. Using these two methods natural frequencies and damping ratios of the first and second modes of the system are extracted and shown in Table 1. Table 1 Exact and estimated natural frequencies and damping ratios of the simulated 4DOF system Mode Number Exact Vales Estimated Values with CFDD Method Estimated Values with MCFDD Method The results show that the modified method results in more accurate modal parameters. Especially, comparison of the estimated damping ratios of the first mode with the exact value shows that the modifications are very effective for the case of coinciding harmonic and natural frequencies Figure 1 SVD graph of the simulated 4DOF system 5

6 db db M.H. Masjedian, M. Keshmiri 4 2 SVD Graph Used Data Mode 1 Used Data Mode 2 Fitted Curve Mode 1 Fitted Curve Mode Figure 2 Elimination of harmonic components and fitted curves for modes 1 and 2 (CFDD method) 4 2 SVD Graph Used Data Mode 1 Used Data Mode 2 Fitted Curve Mode 1 Fitted Curve Mode Figure 3 Elimination of harmonic components and fitted curves for modes 1 and 2 (MCFDD method) 7. EXPERIMENTAL STUDY The effectiveness of the modifications is also evaluated using experimental results on a steel beam. A combination of stochastic and harmonic forces was used to excite the beam. The test setup is shown in figure 4. The responses were measured using 6 accelerometers and an 8-channel vibration analyzer. The harmonic forces were applied with a vibration exciter. The stochastic forces were generated with disordered impacts of fingertips on the beam. The SVD graph for this test is shown in figure 5. Both CFDD and MCFDD methods are used to eliminate the harmonic components in the response. The selected data along with the fitted curves for the third mode are presented in figures 6 and 7 for the two methods, respectively. The natural frequency and damping ratio of the third mode of the beam, extracted by the methods are compared with those extracted from a traditional impact test are all shown in Table 2. The results show that using linear interpolation in CFDD method causes an error in damping ratio for a case that the harmonic frequency is close to the natural frequency. 6

7 db db db 5 th International Operational Modal Analysis Conference, Guimarães May Figure 4 Test setup of the steel beam experiment Figure 5 SVD graph of the steel beam response 8 4 SVD Graph Used Data Fitted Curve Figure 6 Elimination of harmonic components and fitted curves for third mode of steel beam (CFDD method) 8 4 SVD Graph Used Data Fitted Curve Figure 7 Elimination of harmonic components and fitted curves for third mode of steel beam (MCFDD method) 7

8 M.H. Masjedian, M. Keshmiri Table 2 Estimated modal parameters of the third mode of the steel beam Mode Number Impact Test Estimated Values with CFDD Method Estimated Values with MCFDD Method CONCLUSION Most of the OMA methods will fail in the presence of harmonic excitations. Many researchers adapted OMA methods to consider presence of harmonic excitations. CFDD is one of these methods presented recently for this purpose. CFDD uses the kurtosis index to differentiate between the natural frequencies and harmonic frequencies. This paper presented the MCFDD method by two modifications relative to the CFDD method. First, this method uses limited data in the vicinity of each mode instead of all data in the frequency band, to form regression problem. The second modification consists of eliminating the data corresponding to harmonic frequencies in regression problem instead of linear interpolation. The accuracy and efficiency of the CFDD method and the modified method are compared by applying the methods on the response of a numerically 4DOF simulated system and on the response of steel beam experiment setup. The results of extracted natural frequencies and damping ratio showed that eliminating the data corresponding to harmonic frequencies in regression problem in MCFDD method reduces the estimation errors especially in the values of the damping ratio for the case that there is a harmonic coinciding on a structural mode. REFERENCES [1] Brincker R., Andersen P., and Mooller N. (2) An Indicator For Separation of Structural and Harmonic Modes in Output-Only Modal Testing. In: Proceeding of the 18th IMAC [2] Jacobsen N.J. (26) Separating Structural Modes and Harmonic Components in Operational Modal Analysis. In: Proceeding of the 24th IMAC [3] Jacobsen N.J., Andersen P., and Brincker R. (27) Eliminating the Influence of Harmonic Components in Operational Modal Analysis. In: Proceeding of the 25th IMAC [4] Jacobsen N.J., and Andersen P. (28) Curve-Fitted Enhanced Frequency Domain Decomposition- a Robust Technique to Harmonic Excitation in Operational Modal Analysis. In: Proceedings 15th International Congress on Sound and Vibration [5] Brincker R., Zhang L-M., and Anderson P. (2) Modal Identification from Ambient Response using Frequency Domain Decomposition. In: Proceeding of the 18th IMAC [6] Brincker R., Ventura C., and Andersen P. (21) Damping Estimation by Frequency Domain Decomposition. In: Proceeding of the 21st IMAC [7] Andersen P., Brincker R., Venture C., and Cantieni R. (27) Estimating Modal Parameters of Civil Engineering Structures subject to Ambient and Harmonic Excitation. In: Proceeding of the EVACES 7 Conference. 8