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The Observed Wander of the Natural Frequencies in a Structure

¹ Puerto Rico Strong Motion Program
Department of Civil Engineering and Surveying
University of Puerto Rico at Mayaguez
Mayaguez, Puerto Rico 00681
jclinton{at}uprm.edu
jclinton{at}ecf.caltech.edu
 (J.F.C.)

² Department of Civil Engineering
Division of Engineering and Applied Science
California Institute of Technology
Pasadena, California 91125
case{at}caltech.edu
heaton_t{at}caltech.edu
 (S.C.B, T.H.H.)

³ Lockheed Martin Aeronautics Company
Palmdale, California 93599
javier{at}gps.caltech.edu
 (J.F.)

View larger version (34K): [in a new window]   Figure 1. Millikan Library. (a) View from the northeast. The two dark walls in the foreground comprise the 30.5-cm-thick east shear wall, which is somewhat narrower on the ground floor because of walkway openings. The wall panels and concrete moment frame are visible on the north face. (b) North–south section. Walkway openings on the ground floor, which cut through the shear walls, are represented by crosses. (c) Typical plan view. The dark circle is the approximate position of SCSN Station MIK on the ninth floor, the arrows are the approximate positions and orientations of the three USGS channels on each floor (from first floor to roof).

View larger version (33K): [in a new window]   Figure 2. Graphical interpretation of Table 1. Dashed lines are east–west natural frequencies, dashed-dotted lines are north–south-natural frequencies, all from forced vibration testing. Shaded area is the likely region of natural frequencies taking into consideration errors in measurement, caused by unknown shaker weight configuration and weather conditions for each test, and experimental error. Crosses indicate the actual time of a forced vibration measurement. Circles indicate the natural frequency estimated from the strong-motion recording of the event, with the number in italics giving the peak acceleration recorded for the event (cm/sec2).

View larger version (23K): [in a new window]   Figure 3. Graphical interpretation of Table 1—peak rooftop acceleration (accn) amplitude versus frequency, log scaling. For both east–west and north–south, the best-fitting least-squares solution for all the data is plotted in dashed lines. Outlying data from tests and earthquakes prior to main permanent natural frequency shift (pre-Whittier Narrows for north–south; pre-San Fernando for east–west) are removed from dataset for the solid-line regressions, with labeled functional form.

View larger version (36K): [in a new window]   Figure 4. East–west components of CR-1 array in Millikan Library, recorded during the M 6.1 Whittier Narrows earthquake = 19 km, velocity time series. The top trace is from the basement, the second is from the sixth floor, and the last two are from the roof. The last trace includes a sample of how the fundamental frequency of the building is estimated, after the main energy (seen from the basement trace) has passed.

View larger version (56K): [in a new window]   Figure 5. Broad Center. (a) View from the southwest. (b) View from the northwest. Structural core of the unbonded-brace frame is located below the parapet wall visible on the roof. (c) Schematic plan view showing placement of strong-motion sensors. 1 is on the first floor; 2 and 3 are located on the roof.

View larger version (98K): [in a new window]   Figure 6. Spectrogram of natural frequencies as observed at MIK, May 2001 through November 2003. Spectrogram composed of 1-hr time windows, each scaled so maximum is 1 and plotted with linear color bar. Weather data are from JPL weather station ( = 8.5 km). Vertical breaks in data are due to days with data glitches or no recorded data. Tick marks on x axis correspond to first of the month labeled underneath. The peaks in natural frequencies are observed to wander over the course of the 2.5 years. No long-term correlation with temperature is observed, though rain causes temporary lengthening of natural frequency.

View larger version (105K): [in a new window]   Figure 7. Spectrogram of all six horizontal channels at station CBC, the Broad Center, February 2003 through November 2003. One-hour spectrogram windows, no scaling, linear color bar. Sharp red horizontal lines are due to machine noise in the building. Natural frequencies are represented by the broad peaks, near 2.6 Hz, 3.0 Hz, and 3.6 Hz, for the east–west channels, and near 2.4 Hz, 2.8 Hz, and 3.6 Hz for the north–south channels. Notice the 7-day noise cycle with relative quiet on the weekends.

View larger version (43K): [in a new window]   Figure 8. Deviation from the mean natural frequency for the three fundamental frequencies at MIK, May 2001 through November 2003. Fundamental frequencies for each hourly FFT are picked from the peaks in Figure 6, then the deviation from the average is determined. The hourly peak is shown in the thin green line for each frequency; the thick black line tracks the daily average. The thick red horizontal line is the average frequency. Daily rainfall (black), maximum wind gust (green), and maximum temperature (red) from JPL are plotted at the bottom. Vertical blue dashed-dot and red dashed lines indicate days with forced vibration shaking of the library, and earthquakes with motions exceeding 2.5 cm/sec2 at station MIK, respectively, which produce large deviations from the mean (Earthquakes: 9/9/01, M 4.2 Beverly Hills, 7 cm/sec2; 30/10/01, M 6.1 Anza, 2.8 cm/sec2; 3/9/02, M 4.8 Yorba Linda, 5 cm/sec2; 22/2/03, M 5.4 Big Bear, 18 cm/sec2) Note: forced vibration with full weights generates 8 cm/sec2 at north–south fundamental mode.

View larger version (59K): [in a new window]   Figure 9. Deviation from the mean for the natural frequencies of Millikan Library, February 2003, which includes a major rainstorm. In the center of the plot, for each of the three natural frequencies, the horizontal red lines are the monthly average, the black lines are the daily average percent deviation from this mean, and the blue lines are the hourly percent deviation from mean. At the bottom of the figure, the black-bar data are the cumulative hourly rainfall (re-zeros at midnight). The red line is the maximum hourly temperature, and the green is the wind gust. On the top of the figure are each natural frequency amplitude for the hourly FFT peak. The rainfall coincides with a very sharp rise in natural frequencies in the east–west and torsional modes, followed by a slow return toward prerainfall levels. Dashed vertical lines represent the start of each new day (12 a.m. PST). Frequency spikes are due to instrument glitches (6, 21, and 27 February), vibration testing (10 February), and the Big Bear earthquake (22 February). FFT peaks fall at night and on weekends. No major increase in excitation amplitude occurs during rainfall events not associated with high winds.

View larger version (66K): [in a new window]   Figure 10. Spectrogram of fundamental and first overtone for east–west mode during February 2003, a time of heavy rainfall. One-hour spectrogram windows, scaled by the maximum value, linear color bar. JPL weather on bottom. Note correlation between rises in both natural frequencies and heavy rainfall.

View larger version (47K): [in a new window]   Figure 11. As Figure 9. (a) For a 5-day period in January 2003, with a particularly intense wind storm (no rainfall). The natural frequencies of the Library dramatically shorten for the duration of the most intense windstorm, most notably in the east–west direction, but also significantly for the north–south and torsional modes. The smaller windstorm in the evening of January 6 appears slightly shifted in time from the response; this may be due to variations in the winds between the Library and the weather station at JPL. The mean frequencies are for the whole month of January. (b) For a 22- day period, late August to early September 2002. Typical daily variation during the summer months. Note correlation of rise in all three natural frequencies during hottest days. 26 August, vibration testing; 3 September, M 4.8 Yorba Linda Earthquake.

View larger version (40K): [in a new window]   Figure 12. East–west components of USGS-Caltech Array in Millikan Library, M 5.4 Big Bear earthquake: velocity time series and FFT. Includes floors 1–9 and roof of Millikan Library as well as GSA. ("free-field" site) and MIK (on ninth floor of Millikan Library), plotted underneath the first and ninth channels. FFT plots are centered around the fundamental frequency (1.06 Hz) and first overtone (4.55 Hz) for the east–west direction. Also included are the two east–west mode shapes as identified from forced vibrations (Bradford et al., 2004).

View larger version (89K): [in a new window]   Figure 13. Acceleration data and resulting spectrograms for MIK from 22 February 2003. M 5.4 Big Bear earthquake occurs at 12:19 UTC. Maximum acceleration = 19.3 cm/sec2 north–south, 10.9 cm/sec2 east–west. Spectrograms are set around the four natural frequencies. (a) 12:00–13:00 UTC, 30-sec-window spectrograms, with slices shifted by 15 sec. The natural frequencies shorten considerably during the mainshock shaking and aftershocks. Analysis of long-term behavior, especially that of the second mode east–west, is limited by poor resolution due to short FFT length. (b) 11:30–14:30 UTC, 5-min window spectrograms, with slices shifted by 1 min. After shaking has dissipated, over this time window it is clear there is no perceptible long-term shortening of the fundamental frequencies due to the earthquake. The event at 168 min is a M 4.1 aftershock.

View larger version (63K): [in a new window]   Figure 14. Acceleration data and spectrograms from CBC east–west channels, 11:30–14:30 UTC. Five-minute window slices, each shifted by 1 min. There are two spectrograms for each east–west channel; the first is a broad frequency range from acceleration data showing all identified modal frequencies; the second is a frequency band around the first mode using displacement data, which accentuates the behavior of the only mode without nearby machine noise (see Fig. 7). M 5.4 Big Bear earthquake occurs at 12:19 UTC. Maximum acceleration = 14.8 cm/sec2 at BL6, northwest Roof. During the event, the fundamental frequencies shorten considerably. After shaking, over this time window there is no perceptible long-term shortening of the fundamental frequencies due to the earthquake.

View larger version (45K): [in a new window]   Figure 15. Prediction of MIK east–west motions using GSA east–west as input motion. Motions recorded at GSA are convolved with an impulse response at a particular frequency. AMBIENT_SDOF uses the fundamental frequency as determined from FFTs of pre-event data, FORCED_SDOF uses the fundamental frequency as determined from forced vibration tests. (a) M 2.0 18 June 2003 Pasadena earthquake ( = 5 km). A nonyarying second-mode SDOF response is included for both ambient and forced models to include the high-frequency components of the motions. During this very small amplitude motion, the natural frequency determined using the ambient data models the observations at MIK better than the forced vibration result. (b) M 5.4 22 February 2003 Big Bear earthquake ( = 119 km). Neither the AMBIENT_SDOF or FORCED_SDOF response model the MIK observation well. Another model, FFT_SDOF, with natural frequency at 1.06 Hz, as determined by the FFT peak of the strong-motion record is included. This leads to a somewhat improved model of the observed motion. Clearly, even during the moderate shaking generated from this relatively small and distant earthquake, a linear building model is inappropriate. A model that can have evolving natural frequencies will best represent the observed motions.