1. Extracting Dynamic Characteristics from Strong Motion Data Palle Andersen Structural Vibration Solutions A/S Denmark www.svibs.com

2. Contents Introduction Traditional Modal Analysis and Natural Input Modal Analysis Case Study: The 777 Tower in LA, CA Conclusions

3. Introduction Dynamic characterization of civil engineering structures is typically performed by use of ambient excitation (weak motion) The obtained dynamic characteristic are usually the modal parameters. Natural frequency, damping ratio and mode shape

4. Modal Behavior Showing that y the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Displacements y(t) = q 1 (t) 1 + q 2 (t) 2 + q 3 (t) 3 + + q n (t) n = ++ +

5. Introduction The modal parameters are constant for a linear structure. If the structure is non-linear the parameters will change when response levels are changing.

6. Why use strong motion data? Other modes excited than in the case of ambient excitation. To obtain dynamic characteristics describing the strong motion state if a building is non-linear. To gain confidence on FE models used to predict the response of buildings due to strong ground shaking

7. Traditional Modal Technology Input Output Time DomainFrequency Domain FFT Inverse FFT Excitation Response Force Motion Input Output H( ) = == Frequency Response Function Impulse Response Function

8. Combined System Model (analysis procedure) Measured Responses Stationary Zero Mean Gaussian White Noise Model of the combined system is estimated from measured responses Excitation Filter (linear, time-invariant) Structural System (linear, time-invariant) Unknown excitation forces Combined System Modal Model of Structural System extracted from estimated model of Combined System

9. Assumptions Mathematical Stationary input force signals can be approximated by filtered zero mean Gaussian white noise –Signals are completely described by their correlation functions or auto- & cross-spectra –Synthesized correlation functions or auto- & cross- spectra are similar to those obtained from experimental data Practical Broadband excitation All modes must be excited

10. Non-Parametric Modal parameters are estimated directly from curves, functional relationships or tables Parametric Modal parameters are estimated from a parametric model fitted to the signal processed data Identification Techniques Experimental dataSignal Processing Parameter Estimation Parametric Model Modal Parameters Experimental DataSignal Processing Modal Parameters

11. Identification Techniques Non-parametric method: »Frequency Domain Decomposition, FDD Parametric methods: »Enhanced Frequency Domain Decomposition, EFDD »Stochastic Subspace Identification, SSI Signal ProcessingModal Fit Measured Data f n,, FDD (Pure Signal Processing) EFDD (Simplest fit) SSI (Advanced fit) Time Histories

12. Frequency Domain Decomposition (FDD) White noise excitation: Partial fraction expansion: Lightly damped structure: Power Spectral Density (PSD) estimation

13. A number of modes can often not be found by simple peak-picking Modes may be coupled by small frequency difference or by high damping The number of modes equals the number of terms in the linear decomposition in the modal transformation The number of terms is the rank of the PSD matrix The spectra can be used for Operational Deflection Shapes but do not contain modal information! Frequency Domain Decomposition (FDD) Extracting Modal Parameters from PSD response matrix

14. Modal Behavior Showing that y the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Displacements y(t) = q 1 (t) 1 + q 2 (t) 2 + q 3 (t) 3 + + q n (t) n = ++ +

15. Frequency Domain Decomposition (FDD) y(t) = [ ]q(t) [C yy ( )] = E{y(t+ )y(t) T } [G yy ( )] = [ ][G qq ( )] [ ] H [C yy ( )] = E{[ ]q(t+ )q(t) H [ ] H } = [ ][C qq ( )] [ ] H [G yy ( )] = [ V ][S] [ V ] H Correlation: Modal behaviour: => i.e. by Fourier Transform: Same form as Singular Value Decomposition of PSD:

16. Singular Value Decomposition of Hermitian matrices [A] = [V] [S] [V] H = s 1 v 1 v 1 H +s 2 v 2 v 2 H +.. The Singular Value Decomposition of the response matrices is performed for each frequency A real diagonal matrix Number of non-zero elements in the diagonal equals the rank [S] = [V] = Orthogonal columns Unity length columns Approximates the Mode shapes Frequency Domain Decomposition (FDD) Singular valuesSingular vectors

17. Singular Value Decomposition of PSD Matrix Frequency Domain Decomposition (FDD) [G] Frequency [G] i SVD performed for each frequency, response spectra of modes Frequency at peak found from decoupled modes The singular vectors approximates the mode shapes No damping estimated PSD Mag.SVD of PSD S 1 : At least one mode exists S 2 : At least two modes exist Decoupled Modes Mode 1 Mode 2 [A] = [V] [S] [V] H = s 1 v 1 v 1 H +s 2 v 2 v 2 H +..

18. Enhanced FDD Method (EFDD) Estimates Frequency & Damping from each data sets Lists Average Values and Standard deviation Simple curvefitting is used (linear regression)

19. Singular Value Spectral Bell Identification User definable MAC rejection level (default 0.80) »Compare singular vectors i against singular vector 0 Each mode in each data set specified separately 0 i

20. Damping Calculation, 1 Autocorrelation of SDOF Bell using IDFT Graphical feedback of selected interval »maximum and minimum correlation values

21. Damping Calculation, 2 Damping from Logarithmic Envelope of correlation function Logarithmic Decrement method

22. Frequency Calculation From how frequent the correlation function crosses zero

23. Improving and enhancing FDD -> EFDD IFFT performed to calculate Correlation Function of SVD function Frequency and Damping estimated from Correlation Function Mode shape from weighted sum of singular vectors » H = Complex Conjugate transpose (Hermitian) of vector/shape, s1s1 s2s2 0 i MAC = Improved shape estimation from weighted sum: Select MAC rejection level (default 0,8):

24. Case Study: The 777 Tower in LA, CA Left: Figueroa at Wilshire Tower (1990, 52 Storey) Right: The 777 Tower (1989, 54 Storey) Data and pictures are provided by Dr. Carlos E. Ventura, University of British Columbia, Vancouver, BC, Canada. Both building are permanently intrumented by the Califonia Division of Mines and Geology – Strong Motion Instrumentation Program (CSMIP) On January 17, 1994 the instrumentation recorded valuable data of the Northridge earthquake. The simulatenous collected data from the two buildings have been used in a number of comparison cases concerning; base shaking experience, response (shock) spectra, modal characteristics based on strong motion data.

25. Case Study: The 777 Tower in LA, CA Dimensions of footprint: 64.6 x 41.4 m. Overall elevation above ground is 218 m. Has a 4-storey garage below ground. Instrumentation is at: P4 (lowest), ground, 20th, 36th, 46th and penthouse levels.

26. Case Study: The 777 Tower in LA, CA The epicentre of the Northridge earthquake was approx. 32 km from the building. The 20 accelerometers recorded motions for about 180 seconds. Sampling rate 100 Hz. Transducer 18 (Penthouse level). Peak value of displacement is 16.7 cm. The strong ground shaking is only a few seconds, but the response is more than 180 seconds. During the ground shaking the higher modes dominates. After the shaking the lower (1st) mode dominates.

27. Case Study: The 777 Tower in LA, CA Animation og part of the event (~1 min) Unmeasured points animated through linear interpolation. Transducer 18 (Penthouse level). Peak value of displacement is 16.7 cm. The strong ground shaking is only a few seconds, but the response is more than 180 seconds. During the ground shaking the higher modes dominates. After the shaking the lower (1st) mode dominates.

28. Case Study: The 777 Tower in LA, CA First 6 modes identified with Frequency Domain Decomposition (FDD)

29. Case Study: The 777 Tower in LA, CA Mode 1. f = 0.1628 Hz, T = 6.143 Sec. Mode 2. f = 0.1953 Hz, T = 5.12 Sec.

30. Case Study: The 777 Tower in LA, CA Mode 3. f = 0.3662 Hz, T = 2.731 Sec. Mode 4. f = 0.5046 Hz, T = 1.982 Sec.

31. Case Study: The 777 Tower in LA, CA Mode 5. f = 0.5371 Hz, T = 1.862 Sec. Mode 6. f = 0.8219 Hz, T = 1.2167 Sec.

32. Conclusions Modal analysis has been applied to strong records of the Northridge earthquake. 6 lowest modes has been extracted using the Frequency Domain Decomposition (FDD) method. The extracted modes characterizes the dynamics of the investigated building at the strong motion level. In case of non-linear structures, the characteristics will be different from what would be observed in case of a more traditional modal survey using ambient (weak motion) excitation.