# Low-Frequency Vibrations

#### Detection with High-Rate Data and Filtering

*By Ana P. C. Larocca, Ricardo E. Schaal, and Augusto C. B. Barbosa, University of São Paulo*

**Multipath makes it difficult to detect very low-frequency structural vibrations, ranging from 0.05 to 1 Hz, important in characterizing dynamic loads and determining safe structural lifetimes. The authors have developed a phase-residual method for use with very high-frequency data to distinguish receiver noise, multipath, and the periodic displacements that are most structurally significant. The methodology can apply to bridges, tall buildings, and towers.**

Civil engineers continuously seek reliable methods and tools to improve the quality and lifetime of large structures. Most studies in this field have been based on static loading. Nowadays, dynamic loading has become a particular concern, and GPS offers direct measures of dynamic displacements of large structures induced by traffic, wind, and earthquakes.

Precisely characterizing the vibrations that are a common behavior of large structures such as bridges, tall buildings, and towers undergoing dynamic loads facilitates structural analysis studies. It is feasible to detect structural vibrations using a computational model and GPS sensors. The critical vibration frequencies of bridges detectable with different GPS positioning techniques (real-time kinematic, static, quasi-static) range from 0 to 0.3 Hz.

However, the unavoidable presence of multipath signals in the same frequency range makes it difficult to detect very low-frequency vibrations, mostly ranging from 0.05 up to 1 Hz, for short- to medium-span bridges.

Our preliminary results show that the structural vibration measurements, mixed with random amplitude and frequency signals generated by electronics and the ionosphere, together with slowly varying signals generated by multipath, can be better detected with an oversampled GPS data set. This hypothesis relies on fact that the structure oscillation is reasonably stable during the data-collecting period.

The analyses of GPS time series used were done by mathematical addition of well-known sine waves in the raw phase of a 100-Hz data set collected from a short baseline. This strategy simulates the antenna vibrating vertically on a structure, for example at the deck’s midpoint of a bridge.

### Methodology

The methodology used to collect and analyze GPS data was developed for providing low-cost high-accuracy monitoring with single-frequency GPS receivers. The technique is the interferometry method based on the analysis of the L1 double-difference phase residuals of regular static observations. In this data-processing, one satellite is considered as a reference, and its selection is according to the direction of the vibration to be measured. The satellite not taken as a reference — located in the same direction as the vibration movement — has the residual values that contain information about bridge deckvibrations (phase changes). In 2001, we named this the phase-residual method (PRM); see “Millimeters in Motion” in *GPS World*, January 2005.

The residuals incorporate all phase deviations from the adjusted double-difference position during the observation. These phase deviations are due to electronic receiver noise, multipath, small dynamic antenna movements, and other error sources. Converting the residuals to the frequency domain by the fast Fourier transform (FFT) associated with a continuous wavelet transform (CWT), it is possible to see the different behaviors of the receiver phase noise,

multipath, and periodic vibration, enabling the distinction between them. The periodic displacement presents a peak due to the fundamental vibration mode, while the receiver noise presents a white-noise spectrum, and the multipath presents a broad spectrum close to zero frequency. The last feature is very dependent on how the antennas “see” their vicinity. As PRM does not need well-known coordinates epoch-by-epoch to determine the amplitude and the frequency values of the oscillations, it is possible to get reliability.

The spectrum analyses were done by FFT, which provides a design of the vibration’s peak amplitude values; the CWT was used to detect the variation of the frequency value during the timespan of observations, and for validating the results.

### Simulation and Filtering

The preliminary investigation was done by the mathematical addition of sine waves on satellite signals close to zenith, which are the most affected by a vertical amplitude vibration in a real situation. The double-difference phase was calculated, taking as reference the lowest satellite.

The mathematically generated sine wave had peak-to-peak amplitude of 1 millimeter and frequency values ranging from 0.06 Hz up to 1 Hz. The analyses for sine-wave detection were done by applying the FFT and the CWT with the Morlet Wavelet, which deserves a short description.

The CWT was used because structural vibration signals with small peak-to-peak amplitudes in the low frequency region are not well represented in time and frequency by the FFT methods. A particular wavelet, Morlet, was used and is defined as

where w* _{o}* is dimensionless frequency and η is dimensionless time. When using wavelets for feature extraction purposes, the Morlet wavelet is a good choice, because it provides a good balance between time and frequency localization.

The idea behind the CWT is to apply the wavelet as a band-pass filter to the time series. The CWT of a time series (*f* (*t*),*t* = 1,…,*N*) with uniform time steps *dt*, is defined as the convolution of *f* (*t*) with the complex combination of the mother wavelet scaled and normalized, as:

where *W _{j,k}(t)* represents the similarity between wavelet function and the analyzed time series

*f*(

*t*); that is, the higher the value of

*W*, the greater the similarity between the analyzed function and the mother wavelet function that modulates the analyzed signal. The CWT was implemented in MATLAB software.

_{j,k}(t)### 100-Hz Phase Data

Regarding the detection of low frequencies due to a small peak-to-peak amplitude vibration, it is important to show the L1 double-difference residuals of a 100-Hz data rate (Figure 1) and its spectrum before mathematically adding the sine-wave signal due to periodic vibrations. The figure shows the raw phase residuals of 20 seconds of data between two satellites, SV05 (lowest) and SV20 (highest).

Figure 2 presents a 1-second data span for better visualization of peak-to-peak amplitude of the raw double-difference phase residuals, which is lower than 3 millimeters.

Figure 3 was produced to verify the variability of 100-Hz residuals and the probability of errors in the signal that can contribute to degrading the identification of the sine-wave vibration peaks. The resulting histogram is close to a bell curve of a Gaussian distribution, demonstrating the good quality of the 100-Hz data. Figure 4 shows the Morlet CWT computed to identify the low-frequency bias term and a high-frequency noise term. The 5-percent significance (95-percent confidence) level of significant signal-wave information is delimited by a thick contour. The signal information of double-difference phase residuals was used as a reference for supporting a better distinction between noise and sine-wave signals.

### Zero-Baseline Test

A zero-baseline test was performed to determine the correct operation of a GPS receiver, associated antennas, and cabling. The objective was to verify the precision of the receiver. A 1-minute data sample was collected. Figure 5 shows the residuals of L1 double-difference phase.

Figure 6 shows 5 seconds of the zero-baseline data; the peak-to-peak amplitude of residuals is very small, close to 2.0 millimeters. This information leads us to expect detection of very low-frequency vibrations, ranging up to 0.3 Hz with a 1-millimeter amplitude displacement peak-to-peak.

Figure 7 shows the spectrum of the zero-baseline residuals; it is possible to observe the region close to zero strongly affected by multipath. This makes the detection of very low frequencies difficult.

The CWT was applied to decomposing the zero-baseline double-differenced residuals into a low-frequency bias term and a low-frequency noise term. Figure 8 shows the behavior of the residuals of the 100-Hz phase data, where red regions represent the most suggestive energy level of the measurement noise term.

### Preliminary Simulation Results

Figure 9 illustrates the raw L1 double-difference phase residuals with a periodic sine wave of 1 millimeter peak-to-peak amplitude mathematically added to the time series. It is possible to observe the presence of the periodic signal.

Figure 10 shows that the stronger energy is close to 1 Hz due to the 1-Hz sine wave, as expected. The resulting well-defined peak is due to the high sampling rate provided by 100-Hz receivers. Figure 11 shows details of the peak due to the sine wave of 1 Hz added to the residuals.

We analyzed these data with the Morlet CWT to find events to compared when other low frequencies had been simulated, helping separate noise from signal. Figure 12 presents the standardized time-series residuals, showing a region with highest power level. The continuous red region corresponds to a 1-Hz sine wave, and the spread-out red-orange regions may be due to electronic noise and multipath. The region outside the cone, delimited by the thick contour, indicates the detection of significant signal information but without the 95-percent confidence.

**0.5-Hz Sine Wave.** The second sine wave generated had the same peak-to-peak amplitude, 1 millimeter, and the frequency value of 0.5 Hz. Figure 13 illustrates the raw L1 double-difference phase residuals with a periodic 0.5-Hz sine wave mathematically added to the time series.

Figure 14 shows an energy peak at a frequency of approximately 0.5 Hz, also with a well defined peak.

Figure 15 shows details of the peak.

The CWT in Figure 16 shows that the intensity energy level represented by the red continuous region and the spread-out red-orange regions are quite similar to those of the CWT of the 1-Hz sine wave (Figure 12). Note a decrease in energy intensity (orange-yellow) that occurs due to decreased signal sampling of the 0.5-Hz signal (10 cycles) in 20 seconds of data, compared to 1 Hz (12 cycles) in the same 20 seconds.

**0.1-Hz Sine Wave.** The third sine wave mathematically generated had the same peak-to-peak amplitude, 1 millimeter, and a frequency of 0.1 Hz. Figure 17 illustrates the raw L1 double-difference phase residuals with the periodic 0.1-Hz sine wave mathematically added to the time series. Figure 18 shows the power at one frequency, approximately 0.10 Hz, still with a well-defined peak.

Figure 19 presents identification of the 0.1-Hz sine wave by CWT with the 5-percent significance level shown as a thick contour. A decrease of energy intensity (orange-yellow) occurs due to decreased signal sampling of 0.1 Hz (2.5 cycles) in 20 seconds of data compared to 0.5 Hz (10 cycles) in the same 20 seconds.

**0.08-Hz Sine Wave. **We simulated a sine wave of this frequency (Figure 20). Figure 21 presents identification of the 0.08-Hz sine wave by CWT through the 5-percent significance level shown as a thick contour. A decrease in energy intensity (orange-yellow) occurs due to decreased signal sampling of 0.08 Hz (almost two cycles) in 20 seconds of data compared to 0.5 Hz (ten cycles) in the same 20 seconds.

**0.06-Hz Sine Wave.** Finally, a 0.06-Hz sine wave was simulated and added to the residuals, but the FFT spectral analysis did not present the power peak. This can be attributed due to the sine-wave period providing only 1.5 cycles during 20 seconds and did not generate enough power to be detected by FFT.

Figure 22 presents a close-up view of 0.06-Hz sine-wave power spectrum of the residuals not indicating a significant peak close to the expected frequency region.

The investigation continued with a Morlet CWT. In Figure 23 it is possible to verify the presence of a faded red region close to the period corresponding to 0.06 Hz — at the bottom of figure and under the cone’s thick contour — signalling that the wavelet was able to detect a very low frequency even with a small sampling. However, due to small signal sampling, the detection is not within a 95-percent confidence. Otherwise, if the time series had lasted more than 20 seconds, certainly the sine wave would have been detected.

These analyses suggest that longer time-series data would enable detection of very low frequencies with 95-percent confidence.

### Conclusions

The lack of amplitude accuracy does not constitute a significant restriction in large structure monitoring, as the exactness of its natural oscillating frequency, harmonics, and response to external dynamic forces are more important for identification of a structural problem.

Using 100-Hz receivers to detect very low-frequency vibrations, the combination of 100-Hz data with filtering techiniques enables detection of signal vibrations of very low frequencies. The tests were conducted using a mathematical simulation of sine waves added to raw residuals of L1 double-difference phase.

The results of simulations and filtering techniques indicate that very low frequency vibrations can be detected when the sampling rate of GPS data and the sampling frequency of an embedded sine wave is large.

Additionally, zero baseline and static short baseline trials have been conducted to assess the noise of the receivers that is close to 2.5 millimeters — extremely low and contributing to detection of vibrations with low peak-to-peak amplitude.

Spectral analysis is a fundamental tool for engineering development. Despite such new analysis concepts as FFT and CWT used here, as well as higher-order spectra, basic frequency domain analysis will remain the practical analysis tool in the foreseeable future.

Future tests will be carried out collecting 100-Hz data, sufficient for having oversampling of sine-wave frequencies due to structural vibrations, and using a new methodology with just one GPS receiver.

### Acknowledgments

Thanks to the JAVAD GNSS Moscow Research and Development team for providing a Triumph receiver and 100-Hz data through Michael Glutting, whom we also thank. The researchers received a sponsorship from the National Counsel of Technological and Scientific Development Government (CNPq) of the Brazil Federal Government to purchase a pair of 100-Hz data-rate GPS receivers.

### Manufacturers

The 20 seconds of data were kindly provided by JAVAD GNSS Moscow Research and Development team and were collected using Javad GNSS Triumph receivers with JNS choke-ring antennas.

*Ana P.C. LaRocca is a lecturer in the Department of Transportation Engineering of the Polytechnic School at the University of São Paulo (USP) and holds a Ph.D from that same institution.*

*Ricardo E. Schaal is an associate professor with a Ph.D. from USP.*

*Augusto C. B. Barbosa is a Ph.D candidate at the Institute of Astronomy, Geophysics and Atmospheric Sciences, at USP.*