Method

The building’s resonant frequencies were determined by performing a frequency sweep of the building (Resonant Amplification Method, Clough and Penzien (1993)), for which 0.01 Hz frequency increments were used. A plot of roof displacement (normal- ized by the excitation force) versus frequency is created, and the frequency with the largest normalized displacement corresponds to the structure’s resonant frequency. A sample plot is provided in A. It should be noted that I am “resonant frequency” as it is commonly defined, namely, the frequency corresponding to the peak of the resonant amplification curve, which if determined using displacement is found by

ωD =ω0 p

1−2ξ2

where ωD is the resonant frequency, ω0 = p

k/mis the undamped natural frequency of the structure, and ξ is the damping ratio. The resonant frequencies determined by this method are used as initial estimates for the experiment, and have been well documented for Millikan Library, most recently by this thesis, Clinton et al. (2004), and Bradford et al. (2004). However, to determine the building’s resonant frequency at the time of testing, I set up a Kinemetrics Ranger seismometer (1 second pendu- lum, velocity transducer) connected to a readily visible analog chart recorder on the

library’s roof. By monitoring the roof velocity as we vary the excitation frequency, we can find the frequency that corresponds to the building’s maximum roof veloc- ity. This frequency is slightly smaller than ωD, but the difference between the two

frequencies is less than 2% (Kuroiwa, 1969). However, since I do not correct for forc- ing while performing the experiment (the shaker force is proportional to the square of the forcing frequency), the frequency found during the experiment is close to the resonant frequency. All these frequencies are very closely spaced, and they can all be used interchangeably. Therefore, for the duration of this chapter, when I refer to the natural or resonant frequency of the structure, I refer to the resonant frequency found experimentally, using velocity measurements and without correcting for the applied force on the roof top.

The small amplitude of the generated waves at the large observation distances is often below the time domain noise level of the instruments recording the signal. Thus, the data was analyzed in the frequency domain by performing a fast Fourier transform (FFT) for a long time window during the building’s forced excitation. Although the data should ideally be a monochromatic signal, Figure 6.1 shows that the FFT amplitude is not a delta function. Instead, the energy is spread over a narrow frequency band, which indicates that there are small changes in frequency with time. This broadening of the peak is probably due to the feedback mechanism in the shaker’s controller. However, the width of the signal in the frequency domain is less than 0.006 Hz and most of the energy is usually concentrated in a band narrower than 0.002 Hz. This means that the signal is very narrow band and well controlled.

As a result of the signal’s finite frequency content, an integration can be performed in the frequency domain around the frequency of excitation, to account for all of the energy radiated by the building for the duration of the shake. As Figure 6.1 qualitatively shows, the FFT amplitude spectra at all distances are similar in shape once the noise level is considered. I’ll refer to this narrow frequency band containing most of the energy as the radiated spectral band (RSB) and the amplitude at each station of the FFT in this same band as the radiated spectra. The similarity in the radiated spectra is especially clear for stations MIK and PAS. Station MIK has a

three component Kinemetrics Episensor accelerometer located on the East side of the 9th floor of Millikan Library(Clinton et al., 2004), and this data is used as the source spectrum and to determine the RSB. Station PAS contains an ST S −1 broadband sensor, and it is located at a distance of 4.43 km WNW of the source.

MIK North Comp. Source

PAS Radial Comp. Dist. = 4.43 Km GSC Radial Comp. Dist. = 177.05 Km Frequency (Hz) 1.6388 FF T S pec tr al A mp litu de ( m ic ron s) 1.6376 1.6438 1.6314 noise 1.6364 noise radiated spectra frequency of excitation

Figure 6.1 Spectral amplitudes at three TriNet stations: Mil- likan (MIK), Pasadena (PAS), and Goldstone (GSC). The data was recorded during a four hour forced vibration of Millikan Li- brary for shake NS1, and the frequency of excitation was near 1.638 Hz. Notice the similarity in the three Fourier amplitude spectra, especially those at MIK and PAS, which have a very high signal-to-noise ratio. Station MIK is located on the 9th floor of Millikan Library, the source of the radiated signal.

To compute the time domain peak amplitude of the sinusoid generated by the forced vibration of Millikan Library at each station, I integrate the radiated spectra. However, it is necessary to remove the background spectral amplitude noise level from the station spectra, since signals at most of the stations are characterized by very low signal-to-noise levels. This procedure minimizes the influence of the variations in noise levels at the individual stations.

The integration is performed by utilizing a process similar to that used to generate a spectrogram. Starting a significant frequency interval away from the RSB, the FFT spectra is integrated over a moving frequency window as wide as the RSB. By performing an inverse FFT, this integral value is then transformed back into the time domain and stored in a vector, the frequency window moves by one sample, and the process is repeated until a significant frequency interval is achieved on both sides of the RSB. The individual time domain displacement values (from the IFFTs) are assigned the central frequency values of the integral window to track the locations of the original frequency values.

The background spectral noise level is determined from the frequencies immedi- ately adjacent to and extending 0.05 Hz from the radiated spectra. The mean value for each of these noise ranges is determined, and both values are averaged to estimate the noise level for the radiated spectra,N. The chosen frequency range of 0.05 Hz is sufficiently broad for calculating the spectral amplitude noise level, as this frequency window is 20 times wider than the RSB. The mean standard deviation of the indi- vidual time domain values for these same frequency bands is considered to be the measurement error for the computed displacements, σ. Furthermore, the peak of the time domain values over the RSB corresponds to be the total signal amplitude, AT. Therefore, to find the radiated signal amplitude, AR, the noise level should be

subtracted from the total signal amplitude as follows: AR =

q

A2

T −N2±σ

This method is used to compute the amplitude of the harmonic, monochromatic source signal, AR, at all of the TriNet stations where the building’s excitation was

observed, and only these stations will be presented in the data analysis.