TIME-DOMAIN OUTPUT ONLY MODAL PARAMETER EXTRACTION AND ITS APPLICATION

Transcription

1 IME-DOMAIN OUPU ONLY MODAL PARAMEER EXRACION AND IS APPLICAION Hong Guan, University of California, San Diego, U.S.A. Vistasp M. Karbhari*, University of California, San Diego, U.S.A. Charles S. Sikorsky, California Department of ransportation, U.S.A. * Corresponding Author. Mailing Address: Department of Structural Engineering; University of California San Diego; Building 409 University Center; La Jolla, CA , USA Abstract vkarbhari@ucsd.edu el : (01) (858) Output only modal parameter extraction techniques have significant advantages over their inputoutput-based counterparts in the situations where the input measurements are not available or where precise control of excitations is not possible. With output only methods, ambient excitation can be used to excite the structure and the structure can remain in its operational condition during the test a common requirement for long-term Structural Health Monitoring system. In this paper, the implementation of a time-domain output only modal parameter extraction technique in an Vibration-based Structural Health Monitoring system for a FRP bridge structure is discussed. he ime Domain Filtering (DF) method utilizes the time domain response of the structure due to ambient excitation to compute structural natural frequencies and mode shapes. he technique is applied both to numerical and experimental data. he modal parameter results are compared with other modal parameter extraction techniques, and the applicability of the method to health monitoring of a FRP bridge on California State Route 86 is investigated. 1 Introduction Vibration-based Structural Health Monitoring refers to the practice of using the global dynamic response of structures, due to ambient or forced excitations, to monitor and evaluate the performance of the structure (Ref [1]). A crucial step in the implementation of Vibration-based Structural Health Monitoring systems is the estimation of structural modal parameters, which must be performed regularly to provide input for damage detection or other following steps. For most transportation life-line civil engineering structures, an additional requirement is that the structure must remain operational while such estimation is performed. Ambient modal analysis or Output Only modal analysis is a natural candidate for such tasks because no precise knowledge of the excitation is needed. In this paper the theory and implementation of an output-only modal parameter estimation technique is described. he technique, named as ime Domain Filtering (DF) method, is an efficient algorithm for rapid extraction of modal parameter estimations from ambient excitation data. For validation purposes, modal tests were performed on a series of aluminium beams. he problem was analyzed both using DF technique and other existing techniques. he theory and implementation of DF technique, as well as modal parameter estimation results and comparison with other algorithms is presented later

2 in this paper. In the last section, the applicability of the DF in a Vibration-based Structural Health Monitoring system as applied to a bridge in operation on a heavily travelled state highway is discussed. 2 Background he vibration response of a linear time-invariant dynamic system can be expressed in terms of superposition of modes. In time domain, the output of a linear dynamic system can be expressed in terms of its mode shapes and generalized coordinate as uxt (, ) = φ ( xq ) ( t) r r= 1 where φ r is the rth mode shape and q r is the corresponding generalized coordinates. Assuming all modes are well separated, by applying a bandpass filter to the system response, it is possible to isolate the individual modal components in the response time-history (Ref [2]). r u ( x, t) = φ ( x) q ( t) (2) n n n If, for example, the measured response quantity is acceleration response, the measured acceleration time history due to this modal component, expressed in matrix form, will then be: r r r U (3) n = φnq && n u&& 1n(1) L u&& 1n( N) φ1 n q L q&& (4) = M O M = M && n(1) n( N)] u (1) ( ) pn upn N && K && φ pn r r r r r r r r E U U = φ q&& q&& φ = φ Qφ = Qφ φ, where Q n is a scalar. Or in [ U ] [ n and its autocorrelation [ ] [ ][ ] matrix form: [ ] n n n n n n n φ φ n n n n n n φ φ φ φ 1n 1n 1n 2n 1n pn φ1 n φ2nφ 1n n n φ1n φ O M = M L pn = n E Q Q M M φ pn φpnφ1 n L L φpnφ1 n Where [E n ] is a pxp symmetric matrix of rank 1. A close examination of the structure of the [E n ] matrix reveals that each column of [E n ] is a proportional to the modal vector of the nth mode. Alternatively, he Spectral Decomposition heorem (Ref [3]) states that, a symmetry matrix A can be expanded by its eigenvalues and eigenvectors, λ 0 r u r r A PDP [ u1 u n] r r r r r r = = L O M = λ1u 1u1 + λ2u2u2 + L+λnunun 0 λ n r un If there is no noise in the signal, a spectral decomposition of [E n ] will generate a single non-zero eigenvalue λ 1, and the corresponding eigenvector will be the modal vector φ n. Note that Eq (5) L (1) (5) (6)

3 holds no matter what kind of motion the system is experiencing, either free vibration or forced vibration due to some external excitations. he general steps of DF method start with identifying the frequency region where a certain mode might be located, typically from power spectrum plots of response signal or from the Frequency Response Function if input is measured. If excitation is not measured, sometimes it is difficult to identify regions where certain modes might be located just by inspecting the power spectrum plots. However, as will be shown below, for situations where the excitation is mainly an impact, or multiple impacts, power spectrum of the system response almost always give good indication of mode location. he second step is applying a band-pass filter to isolate the desired modes while eliminate the contribution from other modes. And lastly the matrix [E n ] can be formed and the modal vector can be conveniently extracted. 3 Experimental Validation o evaluate the performance of the proposed DF technique, a series of modal tests were performed on small scale specimens in the laboratory. he specimen consist a series of aluminium beams with various supporting conditions. Different cuts and notches were introduced to some of the beams to simulate the effect of damage. he following discussion will be limited to tests that are of concern to the validation of the DF technique. 3.1 Experiment Setup he aluminium beams tested have a length of mm (36 in) and a width of 76.2 mm (3 in). he height of the beam is 6.35 mm (1/4 in). he beam was setup in such a way in order to simulate a simply-simply supported boundary condition. Special support fixture was used on both ends of the beam to ensure close approximation of the idealized boundary condition was achieved (Figure 1). he center-to-center span of the beam is mm (34 in). Figure 2 shows the data acquisition system which consist of a National Instruments SCXI-1000 signal conditioning module, a DAQPad 6052E data acquisition pad and a laptop computer for data display and storage. Nine PCB 3701G2FA3G ICP accelerometer was mounted on the top surface of the beam, shown in Figure 3. Figure 1 est Setup Figure 2 DAQ System A PCB 086C03 impact hammer was used to excite the structure. During the test, special care was taken so that the hammer applied multiple impacts to the beam (Figure 4). It is considered that this

4 type of excitation is close in nature to the impacts of fast-travelling vehicle applied on a short span bridge when the roadway is not perfectly smooth mm (34 in) Figure 3 Accelerometer Setup Figure 4 Impact Force Figure 5 Power Spectrum of Impact Force 3.2 Data Analysis Anti-aliasing filter was applied to all measured response to minimize the effect of aliasing. A boxcar force window was applied on the input force measure to reduce the noise from DAQ system. Because all the transient response was captured in the measurement window, leakage is not a concern and no windowing function was applied. Frequency Response Functions were then calculated. Average from multiple tests was used for modal parameter estimation using peakpicking and circle-fitting methods (Ref [4]). An example of FRF measurement was shown in Figure 6. A very high coherence throughout the frequency range showed the contamination from noise was minimal. Estimated Mode Shapes and Frequencies of the first 2 modes were also shown in Figure 6. For the implementation of DF methods, no knowledge of input should be assumed. he possible frequency range was identified through power spectrum of system responses, shown in Figure 7. As can be seen from the graph, for impact-type input, the peaks in response power spectrum corresponds with the peaks in FRF and thus to system poles. After the possible frequency range was identified, matrix E can be formed. Modal frequencies and mode shapes were then readily extracted. A comparison of results between DF and FF methods shows very good correlation (Figure 8 and able 1), except for one location of mode 2, which is the node point of that mode and thus prone to numerical errors.

5 Figure 6(a) ypical Frequency Response Functions Mode 1 : Hz Mode 2 : Hz Figure 6(b) Estimated Modal Parameters from FRF Figure 7 Response Power Spectrum

6 Figure 8 Mode Shape Comparison (DF vs FRF) FRF Mode 1 FRF Mode 2 Freq (Hz) DF Mode e DF Mode e Application to SHM System Freq (Hz) / able 1 MAC between FF and DF Results With DF verified against proven modal parameter estimation methods, the algorithm was integrated into a Vibration-based Structural Health Monitoring system installed on a FRP composite bridge in California, United States (Ref [5]). he Kings Stormwater Channel Bridge is a FRP continuous bridge with two equal span of 33 ft each. FRP composite deck supported by carbon shell girders formed the load-carrying system of the bridge. he SHM installed on the bridge incorporated various sensors and wireless data transmission technique. For the purpose of this paper, only related parts of the system were discussed. he vibration response of the structure was collected with a network of 63 accelerometers. Of which, 42 measure acceleration in vertical direction and 21 in horizontal direction. Figure 8 shows the distribution of accelerometers on the bridge. A typical time history, with excitation predominantly provided by passing traffic, is shown in Figure 9. he exact magnitude of the excitation was not measured. However, it has been shown that (Ref [6]), for short-span bridges like this one, the dynamic loading that the vehicle applied on the structure is dominated by the 'axle hop' component, which is similar to a impact force with most of the power spectrum at 8-15Hz frequency range. his frequency range matches with the frequency of the first two vibration modes of the bridge calculated by Finite Element method. hus, the response of the structure was expected to be dominated by the contribution of the first two modes.

7 Figure 8 Accelerometer Location Figure 9 Acceleration Response Figure 10(a) 1 st Mode Mode Shapes (Freq = 9.18 Hz)

8 Figure 10(b) 2 nd Mode Mode Shapes (Freq = Hz) he modal parameter estimation results using DF method together with the base line modal parameters from a forced vibration test carried out shortly after the bridge was built are presented in Figure Conclusion ime Domain Filtering technique was proved to give accurate estimate of modal parameters for systems with unknown inputs. he method is simple and efficient, with the exception of the identification of frequency range, the whole process can be easily automated. hus the method has great potential to be integrated in long-term SHM system applicable to civil structures. he frequency range can be easily detected using an algorithm in the case of less-noisy signals. In the case of noisy signals, frequency range need only be set once and the system fully automated thereafter. Further research needs to be directed to solve the problem of closely spaced modes. he effect of non-impact, non-white-noise input also need to be considered. 6 References [1] Guan, H., Karbhari, V. M., : Remote Structural Health Monitoring of a FRP Composite Bridge, SAMPE 2004 echnical Symposium, May 16-20, 2004, Long Beach, CA. [2] Kim, B.-H., Stubbs, N., and Sikorsky, C. "Local Damage Detection Using Incomplete Modal Data." 20th IMAC, [3] Lay, D. C., "Linear Algebra and Its Applications, 2 nd Ed", Addison-Wesley, [4] Ewins, D. J., "Modal esting: heory and Practice", Research Studies Press, [5] Guan, H., Karbhari, V. M., Sikorsky, C. S., "Health Monitoring of a FRP Composite Bridge Augmented by Use of Web Based and Wireless echnologies", IABMAS'04, Oct 19-22, 2004, Kyoto, Japan. [6] "Dynamic Interaction Between Vehicles and Infrastrcture Experiment, echnical Report", Organisation for Economic Co-operation and Development, 1998, Paris, France.