Evaluating the performance of distributed approaches for modal identification

Sriram S. Krishnan*a, Zhuoxiong Suna, Ayhan Irfanoglub, Shirley J. Dykea and Guirong Yanc

a School of Mechanical Engineering, Purdue University, West Lafayette, IN USA 47906 b School of Civil Engineering, Purdue University, West Lafayette, IN USA 47906

c School of Engineering, University of Western Sydney, Australia

ABSTRACT

In this paper two modal identification approaches appropriate for use in a distributed computing environment are applied to a full-scale, complex structure. The natural excitation technique (NExT) is used in conjunction with a condensed eigensystem realization algorithm (ERA), and the frequency domain decomposition with peak-picking (FDD-PP) are both applied to sensor data acquired from a 57.5-ft, 10 bay highway sign truss structure. Monte-Carlo simulations are performed on a numerical example to investigate the statistical properties and sensitivity to noise of the two distributed algorithms. Experimental results are provided and discussed.

Keywords: Modal Identification, Distributed Computing, FDD, ERA

1. INTRODUCTION Modal identification is the process of analyzing measurement data from a structure to estimate its modal parameters (i.e. frequencies, mode shapes and modal damping) that determine the dynamics of the structure. Modal identification techniques require that the user define the amount of data used to develop the model, number of sensor locations, frequency domain or time domain analysis, computational cost and complexity. These decisions are made based on the type of data taken, amount of noise and the parameters one is planning to identify. Thus, they are a prerequisite for most damage detection, model updating and model calibration approaches. These various approaches require accurate estimates of natural frequencies and mode shapes of the structure for reliable results.

Presently, a wide variety of input-output modal identification methods are used, whose application is based on estimates of frequency response functions (FRFs) or impulse response functions (IRFs). These methods try to perform some fitting between measured and theoretical functions and employ different optimization procedures and simplification based on system realization1-3. Although techniques based on forced testing approaches using impulsive loads or more general broadband excitations have long been used, these methods are not generally feasible in civil engineering applications due to safety, inability to excite the structure or cost4. Civil structures are mostly disadvantaged by the fact that external excitation forces cannot be applied to excite them for reasons of structural damage and safety. This limitation has led to the development of numerous output-only modal identification schemes. The ambient vibration from external factors like wind, soil vibration or small tremors are commonly of multiple input nature and wide band frequency content, stimulating a significant number of modes of vibration. For simplicity, output-only modal identification methods often assume the excitation input as a zero mean Gaussian white noise.

There are two main groups of output-only modal identification methods: nonparametric methods essentially developed in frequency domain and parametric methods in time domain5. The basic frequency domain method (peak-picking) was subsequently improved by performing a single value decomposition of the matrix of response spectra, so as to obtain power spectral densities of a set of SDOF systems. This method of frequency domain decomposition (FDD) was better detailed and systematized by Brincker et al6. For distributed computing environments, a decentralized method that added a peak-picking stage is employed to look for system-wide modal frequencies from the power spectra estimates7.

The time domain parametric methods involve the use of mathematical model to idealize the dynamic structural behavior

*krishna3@purdue.edu; phone 1 213 400 0833

and the identification of the modal parameters using some criteria to reduce the error from the experimental signal. The eigensystem realization algorithm developed by Juang and Pappa8 based on Ho-Kalman system realization is a popular time domain method for many structural systems. The ERA method produces natural frequencies, damping ratios and mode shapes directly.

In recent years, smart wireless sensors with on-board computing capabilities are gaining momentum in application to structural engineering. With the advances in this technology, to take advantage of the onboard computing platforms, classical output-only modal identification methods have been modified for distributed computing. In the past only frequency domain algorithms have forayed into this domain. Herein a time-domain, parametric method for a distributed computing environment based on the ERA is introduced called the condensed ERA-NExT.

In this paper we compare two modal identification methods, including the condensed ERA-NExT and the FDD with peak-picking, both developed for distributed implementations. The accuracy of the two methods is comparable to that of a wired system where full response signal can be communicated. The power needed to reliably deliver the entire raw sensor dataset to the base station for centralized processing reduces lifetime of the system9. Hence, there is a need to keep the data transfer to a minimum, without loss of accuracy of the estimated modal properties. The condensed ERA-NExT algorithm and the FDD with peak-picking are designed to reduce the data transfer across the network. The FDD-PP requires user selection of the natural frequencies from the FRF plots. Here, an automated natural frequency selection procedure based on the condensed ERA algorithm is developed. A Monte-Carlo simulation is performed on a numerical model to study the sensitivity of the estimated modal parameters to noise and amount of data transferred across the network. Subsequently, experimental validation of the method is performed on data from a full scale highway sign truss.

2. THEORETICAL BACKGROUND 2.1 Condensed ERA - NExT

The use of cross-correlation functions of output response signals for a weakly stationary, broad-band and uncorrelated excitation (ambient vibration) as free response decay of the system was presented with the theoretical background by James et al10. The technique, known as NExT, is based on the fact that the cross-correlation function of the responses of the structure with a reference signal satisfies the homogeneous equation of motion and can be treated as free response data. Once the correlation functions are obtained, the ERA is applied by first forming the Hankel Matrix

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)1()()1()(

)1(

rpkYrkY

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(1)

where, Yj(k) = m× n pulse response matrix at jth time step. For good results, the number of rows should be selected as 10 times the modes to be estimated and the number of rows should be selected 2-3 times the number of columns. The size of the Hankel matrix determines the number of discrete time sample of the correlation function required for the ERA method. This requires the use of p+r data points to construct H(0). For most cases the number of data points required for building the Hankel matrix is a fraction of the full free response and the rest of the correlation function is not used. For distributed computing over wireless network it is important to keep the amount of data transfer over the wireless network to a minimum to save energy. Thus in the implementation proposed here, only the required amount of data points are transferred to calculate the cross-correlation function via direct convolution process. As before, ERA is adopted to determine the modal properties of the system by performing the singular value decomposition (SVD) on H(0). The SVD process gives the decomposition of the Hankel Matrix into two unitary matrices and diagonal matrix. The state space representation for the discrete time system is found using

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) (3)

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The discrete time representation is typically converted to a continuous representation. The modal characteristics are evaluated by determining the eigenvalues of the continuous time state matrix, and the eigenvectors are found using

ψψ AZ ˆ1−= (6) C1−=ψφ (7)

where, Z is the digitalization of the discrete time system matrix, the eigenvalues give the natural frequencies of the system and φ gives the mode shapes. Using a reduced amount of data does introduce more noise in the correlation function, but the estimation of natural frequencies via the ERA-NeXT method is shown to be robust in the presence of noise11. Giraldo et. al.12 presented systematic guidelines to eliminate the computational modes in an automated fashion.

2.2 FDD with peak-picking

Brincker6 proposed the FDD method. Data acquired from each of the sensors is transformed into frequency domain by performing an FFT. The cross spectral density (CSD) functions between every node with each of the other node are obtained by applying the Welsh averaging technique over a number of frames. This averaging is performed after applying a Hanning window for every frame to reduce leakage

TYXfSXY

*)(ˆ = (8)

∑= )(ˆ1)(~ fSNseg

fS XYiXY (9)

Where, X and Y represents the raw spectra of the output response. Generally, a frame size of 1024 or 4096 is chosen depending on the variance present in the signal. So for a fixed T seconds of data, there is a trade-off between segment length (Tr), which controls the bias and number of segments (Nseg), which controls the variance: T=Tr. Nseg. A CSD matrix is constructed at each of the discrete frequencies by assembling the CSD (Sij). A singular value decomposition (SVD) is performed on the CSD matrix at each of the discrete frequencies. The singular value in each singular value matrix is collected to form a vector, and the peaks of the singular value are at the natural frequencies. The mode shapes corresponding to the natural frequencies can be estimated from the first column of the corresponding left SVD matrix.

Herein to achieve energy efficiency by reducing the amount of data transferred, we propose that the data flow in the FDD algorithm is optimized to enable an efficient distributed mapping. The peak-picking is introduced at the Welsh averaging stage. Only the magnitudes at the natural frequencies of the raw spectra from each frame are transmitted to the reference sensor for processing. This enables the evaluation of the cross-spectra only at the selected natural frequencies. The CSD matrices are created only at the peak frequencies and not the entire spectrum, thus reducing the computational cost and network transmission cost significantly. The natural frequencies for the peak picking are obtained from the Condensed ERA introduced earlier. After obtaining the CSD matrices at only a few discrete frequencies, the singular value decomposition is performed and mode shapes are obtained as before.

3. NUMERICAL SIMULATION 3.1 Description of the Model

The distributed approaches discussed above are developed for embedding on smart wireless sensors for automated modal identification. Hence, for purposes of demonstrating the variance and repeatability of the approaches to output noise, a typical truss model with 11 bays is used.

This two dimensional truss is modeled using 26 nodes, 49 beam elements to construct a finite element (FE) model. Cylindrical geometry was assumed for all the members. A FE model of the test truss was developed in MATLAB (see Fig. 1). Fixed conditions for all the DOF were defined at left support to simulate rigid connection and roller support condition was defined at the right support connection to the ground. Finally, mass proportional damping was introduced in the homogeneous equation of motion. The exact values of the first four natural frequencies and corresponding mode shapes of the truss are calculated from the state space realization of the FE model (see Fig. 2).

Figure 1: Numerical model and test strategy

The simulation is performed based on the protocols for the experiments discussed in the next section. Ambient vibration responses were simulated by introducing bandwidth-limited, weakly stationary and normally distributed random inputs acting as vertical ground excitation to the model at bay 4.

Due to restrictions in the number of channels in the data acquisition system, several tests are performed to fully test the structure. The full test of the truss is divided into four separate tests (shown in red box) and a combined test (shown with green markers) which is used to stitch the mode shapes together (see Fig. 3). Data from all five tests perform the condensed ERA-NExT and FDD-PP at the respective reference nodes and the mode shapes are stitched together.

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Figure 2: First 4 numerical mode shapes

3.2 Monte-Carlo simulation

Fifty simulations are performed to facilitate a comparison of the condensed ERA-NexT and FDD-PP with the restrictions imposed on the testing procedures. To facilitate the comparison, a centralized computing architecture, the ERA-NexT is performed by transferring all the data points collected to the reference node for computation. The effect of sensor noise is investigated by varying the signal-to-noise ratios. A zero-mean Gaussian broadband noise is added to the acceleration records from each simulation to simulate the effects of sensor noise. The standard deviation of the noise added is varied from 0 – 10% of that of the signal. The natural frequencies and mode shapes identified from the state space realization are used as the true values. Two measures are used to assess the accuracy of model identification methods- the relative error of the identified frequencies from the true frequencies (P1) and modal assurance criterion (MAC) of the mode shapes (P2). At each mode P1 and P2 are calculated as

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3.3 Discussion of results

The values of the relative frequency (P1) for the first four modes of the simulation are shown in Fig. 3 as a function of noise. Each plot in Fig. 3 shows the 50 estimates of the relative frequencies of the Monte-Carlo simulation. The two proposed distributed algorithm perform comparably with the FDD-PP performing slightly better. The error in the estimated frequency is within 0.5% of the actual frequency. Estimation of the natural frequencies using FDD-PP is relatively insensitive to noise because the frequency bands are provided. The Condensed ERA method performs similarly with increase in noise. An important advantage of the condensed ERA method over FDD method is that it provides similar estimate of the frequencies without the need to rebroadcast the frequency bands to all the sensors.

The MAC values are calculated with respect to the true mode shapes obtained from the state space representation. Fig. 4 shows the MAC (P2) comparison of the distributed algorithms as a function of noise. The FDD-PP performs consistently with increase in noise with a slight reduction in its MAC values from 1 to 0.8.

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Figure 3: Accuracy of identified natural frequencies (P1) as a function of noise level (a) Offline ERA, (b) Condensed ERA- NExT and (c) FDD-PP.

Successful implementation of ERA requires experience and some knowledge about the structure. The implementation of the condensed ERA algorithm requires two parameters to setup, namely the dimensions of the Hankel matrices used and the model order used to truncate the singular value decompositions. The accuracy of the realization depends on these parameters and the amount of noise present. The number of data points transferred depends on the size of the Hankel matrix. Generally, additional computational modes are estimated by setting large Hankel matrices and high modal order and using other numerical methods to eliminate the computational modes. The effect of changing the size of Hankel matrix size is inconsistant to detecting the correct modes. The Computational modes, which are not physical but are present due to numerical noise, are

removed. These artificial modes, typically can be identified as they correspond to singular values with low extended modal amplitude coherence (EMAC) and very high damping ratios. The condensed ERA-NExT does not performwell consistently to evaluate the mode shapes. The accurate elimination of computational modes is important to assure effectiveness of ERA-NexT in automated implementation. This phenomenon is the main cause of high variance in the parameter P2 in the condensed ERA method. Stability diagrams could be employed to alleviate this problem with increased computational costs of evaluating multiple Hankel matrices and tabulating the probable frequencies.

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Figure 4: Accuracy of identified mode shapes (P2) as a function of noise level (a) Offline ERA, (b) Condenser ERA- NExT and (c) FDD-PP.

To improve the condensed ERA method, a test was conducted to study the effect of number of points transferred while all other parameters remain constant. The number of points transferred is increased from 200 to 600, which is still a fraction of the total number of data points collected. Fig. 5 shows that the values of the MAC (P2) for the first four frequencies increases with number of points transferred. This is increase is due to the fact that longer time histories provide better cross correlation averaging and reduces noise in the function. There are still cases of very low MAC numbers because for some tests of the Monte-Carlo simulation the computational modes are detected as true modes. Stability diagrams could be employed to alleviate this problem with increased computational costs of evaluating multiple Hankel matrices.

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Figure 5: Accuracy of identified mode shapes (P2) as a function of number of points for condenser ERA- NExT.

4. EXPERIMENTAL IMPLEMENTATION 4.1 Description of full-scale structure

An experiment to demonstrate the proposed approaches is performed on a full-scale truss at the Robert L. and Terry L. Bowen Laboratory for Large-Scale Civil Engineering Research at Purdue University13. The truss is 17.04m long, 1.83m wide and 1.98m high and has 4 horizontal major cord members welded with vertical, diagonal and transverse minor members. All members have cylindrical a cross-section. The major chord members have a diameter of 0.35m and the minor members have a diameter of 0.12m. The truss is made of aluminum alloy (6061-T6). The truss studied here was previously mounted over interstate I-29 near Sioux City in Iowa as a sign support to display route information14.

4.2 Setup of the experiment

The full-scale truss was configured in the Bowen Laboratory with left ends of the truss simply supported by placing it on metal blocks to simulate pinned joint and right ends of the truss were placed on cylinders to simulate roller supports.

Figure 6: Truss setup

An electro-dynamic shaker (VG-100 from Vibration Test Systems) was used to excite the truss along the vertical direction (see Fig. 6). Uniaxial accelerometers (PCB model 3711, MEMS DC response accelerometers) were used to measure the output acceleration data. Three 4-channel, SigLab dynamic signal analyzers (Spectral Dynamics, Inc.) were used collect the input dynamic force and output acceleration data. The truss was excited in the vertical direction with band-limited white noise to simulate ambient excitation in the field. The generated white noise is first amplified using an amplifier and then the amplified noise signal is imposed on the truss using an electro-dynamic shaker. For each of the test configurations, multiple input excitations were used. The truss was subject to white noise over band widths 0-100 Hz and 0-200 Hz.

4.3 Data acquisition

A 12-channel DAQ system restricted the sensor deployment of the full structure in any one test. Thus, a comprehensive test strategy was developed to fully characterize the front plane of the truss in the vertical and horizontal directions.

Seven tests were performed on the truss to fully characterize the truss. In each of the tests, only a portion of truss was equipped with sensors. One of tests was performed with the sensors attached only on the lower front chord of the truss. The other six tests were conducted sequentially to cover nodes from 1 to 23 with 8 sensors in each test. Fig. 7 shows the sensor deployment for each test. In each test, in addition to the responses at the nodes surrounded by the red box, the response in the vertical direction at node 16 was also measured. In each test, the first 9 channels were used to measure the output acceleration while the 10th channel was used to measure the input dynamic force. The sensor located at node

16 (see figure) was maintained as a reference sensor to finally combine all the tests. The response at node 16 was selected as the reference, because node 16 was close to the shaker and thus the reference signal had a good signal-to-noise ratio. Multiple tests were carried out at different input amplitude and different sampling frequency of the measured responses. Data was collected for 160 seconds at sampling frequencies of 256 Hz and 512 Hz.

Figure 7: Sensor deployment in each test

4.4 Results of experimental tests

The proposed distributed modal identification is performed using data acquired with a wired sensor network deployed on the full scale truss. Results from the numerical simulation of the highway sign truss, showed that the first 8 natural frequencies were within 58 Hz and hence the data was filtered and resample to 128 Hz. The length of the signal was for 160sec.

Unlike numerical studies, the exact modal parameters of the structure are not available in experimental testing. Hence, to compare the performance of the proposed methods, ERA is performed using the full data and results are taken as the true modal properties. Accuracy measures P1 and P2 are used to compare the two distributed algorithms. The implementation of the condensed ERA-NExT and FDD-PP are identical to the numerical study described above. Fig 8 shows the first four modes shapes evaluated using the three methods. The four modes are first bending mode, first twisting mode, second bending mode and second twisting mode detected with sensors in the y direction only along the front plane of the truss. The mode shapes evaluated by condensed ERA have a MAC value of in the range .81 to .95. FDD-PP estimates MAC value of 0.94 to 0.98 for the four modes.

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figure 8: First four estimated mode shapes for the three methods

Table 1: Comparison of results

Model Identification

method

MAC (P2) Relative Frequency (P1) Data transfer

per node Mode1 Mode2 Mode3 Mode4 Mode1 Mode2 Mode3 Mode4

Condensed ERA-NExT 0.867 0.819 0.951 0.854 -0.003 0.244 -0.119 -0.013 198

FDD with Peak picking 0.986 0.949 0.972 0.959 0.001 0.258 -0.110 -0.017 156

5. CONCLUSIONS This paper focuses on the ambient vibration-based modal identification of civil engineering structures while limiting the amount of vibration sensor data transfer. It is desirable with wireless sensors to restrict the amount of data transfer to ensure that the power available is budgeted appropriately. The main purpose of this study is to statistically evaluate and compare two distributed architecture implementations of popular time domain and frequency domain modal identification techniques, namely condensed eigensystem realization algorithm (ERA) and frequency domain decomposition with peak-picking (FDD-PP), in an entirely automated environment. The results indicate that both algorithms perform similarly when sensor noise level is low. However, as the noise in sensor data increases, the accuracy of the condensed ERA degrades. However, a significant advantage of the condensed ERA method is that it evaluates the natural frequencies and mode shapes in a single step and does not require user input of frequency bands. Both approaches are implemented on a full-scale highway truss. Good results are obtained with both methods, keeping the data transfer uniform.

ACKNOWLEDGMENTS

The authors would like to acknowledge support from NSF Grant No. NSF-CMMI-1035748. We thank Prof. Robert Connor at Purdue University and Mr. Michael Todsen at Iowa Department of Transportation for helping acquire the truss used in this study.

REFERENCES

[1] Ibrahim, S., R., Mikulcik, E., C., "A Method for the Direct Identification of Vibration Parameters from the Free Response", The Shock and Vibration Bulletin, 47(4), (1977)

[2] Rodrigues, J., Brincker, R., Andersen, P., "Improvement of Frequency Domain Output-Only Modal Identification from the Application of the Random Decrement Technique", Proc. 23rd Int. Modal Analysis Conference, (2004)

[3] Peeters, B., Roeck, G., D., "Stochastic System Identification for Operational Modal Analysis: A Review", J. Dyn. Sys., Meas., Control; 123(4) , (2001)

[4] Wenzel, H., and Pichler, D., [Ambient vibration monitoring], Wiley, New York, 248 (2005). [5] Cunha, A., Caetano, E., Magalhães, F., Moutinho, C., "From input-output to output-only modal identification

of civil engineering structures", Samco report; (2006) [6] Brincker, R., Zhang, L., Andersen, P., "Modal Identification from Ambient Responses using Frequency Domain

Decomposition", Proc. 18thInt. Modal Analysis Conference, Kissimmee, (2001) [7] Zimmerman, A., T., Shiraishi, M., Swartz, R., A., and Lynch, J., P., "Automated modal parameter estimation by

parallel processing within wireless monitoring systems", J. of Infrastruct. Sys., 14(1), 102-113, (2008) [8] Juang, J., N., Pappa, R., S., "An eigensystem realization algorithm for model parameter identification and

model reduction", AIAA Journal of Guid., Control, and Dyn., 8(5), (1985) [9] Lynch, J., P., and Loh, K., "A Summary Review of Wireless Sensors and Sensor Networks for Structural Health

Monitoring", Shock and Vibration Digest, 38(2), (2006)

[10] James, G., H., Carne, T., G., and Lauffer, J., P., "The natural excitation technique for modal parameter extraction from operating wind turbines.", Rep. No. SAND92-1666 (UC-261), Sandia National Laboratories, Albuquerque, N.M., (1993)

[11] Caicedo J., M., Dyke S., J., Johnson E., A., "Natural excitation technique and eigensystem realization algorithm for phase I of the IASC-ASCE benchmark problem: simulated data", J. of Engg. Mech.; 130(1), (2004)

[12] Giraldo D., Dyke S., J., "Modal identification through ambient vibration: a comparative study", 24th Conference and Exposition on Structural Dynamics (IMAC-XXIV), (2006)

[13] https://engineering.purdue.edu/CE/BOWEN [14] Yan G., Dyke S., J., Irfanoglu A., "Experimental Validation of Damage Detection Approach on a Full Scale

Sign Support Truss". Mech. sys. and signal processing, (under review), (2010)