Dynamic Characteristics of Prototype Structures | Wind Engineering

This article is concerned with measuring the dynamic properties of prototype structures, especially those characteristics which are important for calculating structural response to wind loading.

Introduction to the Dynamic Characteristics of Prototype Structures:

In the design of a tall building or a large bridge, the evaluation of structural response to the design wind load may be important both for serviceability and safety considerations. Therefore it is important that the designer has reliable methods for assessing both the wind loads and the structural characteristics in order to enable the structural response to be calculated.

One way of checking that the methods used for these calculations are adequate, is to measure the characteristics and response of existing structures, and compare the behaviour with that assumed in design. This feedback mechanism will certainly be beneficial to individual designers, and will eventually improve the accuracy of predictions made by the wind engineering profession in general.

However, measuring the characteristics or response of an actual structure may not be a simple task, and the purpose of this paper is to consider what measurements can be made and their likely accuracy. Details of how these characteristics are used in analysis and design will be dealt with in other papers in this volume.

The Building Research Establishment (BRE) has been involved with measuring the dynamic characteristics of prototype structures since the mid 1970’s, and most of the methods described in the paper are those used by BRE.

There are a number of different tests which can be used for various purposes, and these can be divided into two main types of test:

First, there is the forced vibration test which is used to determine the characteristics of the lower frequency modes of vibration of a structure, i.e., frequency, damping, stiffness and mode shape.

Second, there is the ambient test, or monitoring the response of the structure to wind loading, and this can be used to determine some of the characteristics of the structure (frequency, damping and mode shape), and also to determine the structural response to a given wind speed.

However, before describing these tests methods in detail, a number of related points are discussed. These are the accuracy of estimating characteristics at the design stage, the measurement of structural response and simple tests to measure frequency.

Estimating Characteristics at the Design Stage:

In order to evaluate the dynamic response of a structure to wind loads, it is necessary to estimate the modal characteristics of the structure. For each mode of vibration these characteristics are frequency, damping, stiffness and mode shape. The lower frequency modes of vibration will have a significant effect on the calculated response of the structure to the wind; hence, it is useful to consider the expected accuracy in estimating these characteristics at the design stage.

A paper by Ellis (1980) examined the accuracy of estimating the fundamental frequency of buildings. It used frequencies measured on a large number of buildings and also made use of published results where both measured and computer based predicted frequencies were given.

To demonstrate the accuracy of empirical predictors an optimum formula was derived for the data. This showed that for a simple formula f = 46/H, (where f is the fundamental translational frequency and H is the height of the building in metres), errors of ±50%, between measured and predicted frequencies, were not uncommon.

However, when the computer based predictions were examined, their correlation with the measured values was worse than that for the simple empirical method. Intuitively, it would seem that detailed computer models should provide better estimates of the fundamental frequencies of buildings than simple empirical predictors, but since 1980 the authors have seen no real evidence that the accuracy of computer based models has improved significantly. Hence unless there is definite evidence to suggest otherwise, errors of ±50% should be anticipated with estimates of natural frequencies.

The importance of obtaining the correct value of natural frequency can be illustrated by examining the case of the response of a tall building to wind loading. The r.m.s. resonant acceleration is inversely proportional to the natural frequency raised to the power of about 1.35. The response of a tall building is also inversely proportional to the mass of the building and is inversely proportional to the square root of the damping. Littler and Murphy (1992) suggest ±20% and -50%, +100% (i.e., a factor of two) as typical bounds of uncertainty for these parameters respectively.

The accuracy of predicting the characteristics of buildings is discussed, and similar errors are to be expected for other types of structure. However, if a structure is relatively simple and its boundary conditions are well defined, then this is likely to lead to a more accurate structural model. There is no reliable way of calculating damping values and the best estimates of damping are usually obtained by adopting measured values from similar types of structure.

The Measurement of Structural Response:

In the following sections a variety of tests are described, but each type of test requires the measurement of the response of the structure. The equipment used for the measurements must be appropriate for the task and used correctly. To monitor response a variety of transducers, e.g., accelerometers or geophones, can be used, but they need to be selected to cover the frequency and amplitude range examined.

The signals from the transducers may need to be filtered and amplified before being displayed or recorded. Oscilloscopes are normally used for displaying the signal, with the signal either being recorded on a tape recorder, or onto a computer based data acquisition system. If the data are recorded digitally then careful consideration should be given to the sampling rate used and the potential problem of aliasing.

This topic is dealt with in Bendat and Piersol (1986). Digital recording is becoming increasingly common, but analogue recording usually has one advantage in that the same signal can be reanalysed with different parameters if required. Finally it is important to note that each item in the measurement chain (transducers, signal conditioning equipment and the recording device) needs to be calibrated.

Simple Tests to Measure Frequency:

In some circumstances all that may be required from a test is to measure the fundamental frequency of a structure, and this can be a relatively easy task. Most of these tests could be performed by engineers given the right equipment and should yield reasonable values of natural frequency.

Perhaps the simplest way to measure the natural frequency of a structure subjected to wind excitation is to display the signal from a measurement transducer on an oscilloscope and, if the signal resembles a single sine wave, determine the time between successive peaks in the time history.

This gives the period of the motion which is the inverse of the frequency. This method works well when the response is primarily in one mode. Careful positioning of the transducer and possible filtering of the signal may be required in order to minimise interference between translational and torsional modes.

Another fairly simple way of measuring frequency is to make a short recording of the response of the structure to wind loading (a response- time history) and then transform this record from the time to the frequency domain using a fast Fourier transform (FFT) routine to produce an auto-spectrum, or a power spectrum.

The short recording could be of several minutes duration. BRE often use this type of analysis when taking remote measurements using their Laser interferometer. This Laser system provides a method of taking measurements without direct access to the structure, which can be useful in certain circumstances.

Alternatively, for relatively small structures, a simple form of forced vibration could be used, which may only take a few seconds and involves monitoring the structural response to a given impact. Such an impact test could be someone jumping on a structure such as a footbridge or a weight being released from a grandstand roof. The frequency can be derived either from the response-time history, or an auto-spectrum of that record. For a number of structures the decay of vibration may also yield a reasonable estimate of the damping.

Forced Vibration Testing:

Steady-state forced vibration testing is expensive and a test on a single building may take several days; however, if a detailed evaluation of all the characteristics of the lower frequency modes of vibration is desired, it is unrivalled in accuracy by any other test method.

In the following sections the test procedure used by BRE is described in detail:

1. The BRE Test Procedure:

BRE has carried out forced vibration tests on many tall buildings and other structures, primarily to evaluate the characteristics of their fundamental modes of vibration. These tests have a major advantage over ambient tests, because the modal stiffness and modal mass can be measured.

BRE has a system of four exciters, which use eccentric contra-rotating masses to produce a horizontal sinusoidal force, and these have been used for testing buildings and dams. For vertical motion BRE has a much smaller system, which has been used for testing floors and grandstands.

For this type of exciter the force is proportional to the eccentric mass used, its radius from the centre of rotation and the square of the excitation frequency (mrω2). The large system can generate a maximum force of 4.1 kN (single peak) at 1 Hz, and the smaller system 167 N at 1 Hz.

2. Basic Test Methodology – Frequench Sweeps:

The exciter(s) are placed in a suitable location and orientation for exciting the structure in the mode to be investigated. The exciters are set to produce a known force and then the forcing frequency is incremented over the required frequency range whilst the response is monitored.

The response is usually monitored using an accelerometer aligned to measure motion in the appropriate direction. The data obtained are normalised by converting the measured accelerations to the equivalent displacement and then dividing this displacement by the applied force. The plot of normalised response against frequency is termed the response spectrum.

A best-fit curve based on a visco-elastic model is then determined for the response spectrum. The parameters used to determine the best-fit model for each mode are frequency, damping and stiffness. The accuracy of these values can be judged by a visual comparison of the experimental data and the fitted curve.

With this type of forced vibration test, it is important that the exciter can provide sufficient force to produce a measurable response in the structure over the required frequency range, and it is equally important that the system can be controlled accurately and that the frequency control is stable.

The BRE system which is used to test buildings works from 0.3 Hz to 20.0 Hz in increments of 0.001 Hz. The phase of each exciter relative to the frequency of the reference signal of the control is checked, and adjusted as necessary, several hundred times per revolution of the weights.

The phase lag of each exciter is displayed for the operator to show when a stable situation has been obtained, which is especially important when the required frequency has just been altered.

If the testing is conducted when the wind is blowing, then the ambient movement of the structure will affect the tests. This is usually only a problem for very large low-frequency systems, and in these cases testing has to be undertaken in relatively calm conditions.

The final requirement for the testing is that the increments of frequency used are sufficiently small when the response is changing rapidly in amplitude, so that the spectrum will be resolved adequately. This means that a large number of points are required around resonance in order to define the fitted curve. Fig. 1 shows a response spectrum obtained for a fundamental translational mode from a test on one tall building, Habersnon Court.

With the facility for changing the applied force, the force vibration testing procedure is also used to examine the characteristics at various amplitudes of vibration. This has shown that the structural characteristics vary with amplitude, and for the range which is of interest to wind engineers, these changes are a small decrease in frequency and a small increase in damping with increasing amplitude of motion. The visco-elastic model also shows a slight skew fit with the measured data, which is another indication of non-linear response, but for wind engineering purposes this is not important and can be disregarded.

Mode shapes can also be determined by setting the exciters to provide a steady state motion at a natural frequency and monitoring the response at various locations throughout the structure relative to a reference accelerometer. This can be used to examine support conditions which may be important for calibrating any detailed mathematical model of the system.

3. Damping From Decays:

The other method of obtaining damping from steady-state testing is by analysing a decay of oscillation. The method now used by BRE is as follows. The exciters are set to produce steady-state motion at a natural frequency and then switched off simultaneously. The decay of oscillation is recorded, usually at a sampling frequency of 1000 Hz for tests on a tall building. If necessary, filtering of the signal can be carried out prior to recording.

The upper part of Fig. 2 shows the decay obtained for the same mode and at the same force level as for the response spectrum given in Fig. 1.

The analysis is then carried out on a chosen part of this decay. The centre part of Fig. 2 shows the portion of the decay from 95% to 35% of the original steady-state amplitude.

A best-fit decay, based on a visco-elastic model, is then derived for the selected part of the decay using a least squares fitting procedure. All the points in the selected part of the decay are used, not just the peak amplitudes. The program uses frequency, damping and initial amplitude values (based on the selected number of cycles and the initial and final values from the chosen part of the decay) as the initial values for deriving the best-fit theoretical decay.

Frequency and damping are considered to have constant values throughout the theoretical decay. The lower part of Fig. 2 shows the selected part of the decay with the best- fit theoretical curve superimposed. It can be seen that the fit is excellent except at the end where some misalignment occurs.

As higher weighting is applied to the points with larger amplitudes, any mismatch is likely to occur at the end of the decay, and this mismatch is generally because of the non-linear response of the structure.

In this case, the damping values obtained by these two entirely separate methods (i.e., from the response spectrum and from the decay) are 1.75% and 1.88%, which for damping measurements can be considered to agree quite well. For tests using exciters whose stability or control is not adequate for defining the response spectrum adequately, damping values taken from decays for fundamental modes of vibration will still be acceptable, as they are less prone to experimental errors.

The two methods for deriving damping are complementary and, where possible, both should be used. Any differences between the results obtained by the two methods which cannot be explained easily as modal interference etc. will show any weaknesses in experimental technique or the mathematical model used, whereas similar results from the two methods will add to the confidence that can be placed in the results.

4. Other Methods of Forced Vibration Testing:

Other methods of forced vibration testing have been used to obtain measurements on tall buildings and similar structures. Daniels et al (1993) used both steady-state and instrumented hammer tests on a two storey building. They state that both methods have the same accuracy but that the hammer tests are considerably quicker and cheaper.

However, the frequency steps used in their steady-state tests (1 Hz) are too coarse for a structure with natural frequencies of 7 to 8 Hz, as was the case in the tests reported. Hammer tests, in which both the load and the response are monitored, can be used for deriving structural characteristics, and for small structures have proved very useful.

However, with larger structures it may be difficult to introduce sufficient energy over the frequency range of interest, thus creating difficulties with response measurements and thereby reducing the accuracy of any derived transfer functions.

Williams and Tsang (1989) used a linear motion hydraulic exciter which could apply wide band random forces to a tall building. They acknowledge however, that there are problems getting sufficient frequency resolution with such exciters on structures with low natural frequencies, and that steady-state sinusoidal tests can generally provide better results.

Spectral Analysis of Ambient Recordings:

The forced vibration tests described in the previous section, introduced a controlled force to excite the structure and thus provide a deterministic test method. But the wind also causes structural vibrations, and the analysis of these vibrations can be used to determine structural characteristics.

Analysing the wind response of a structure does not require the specialist equipment to shake the structure used in forced vibration tests, and monitoring wind response will also show how the structure moves under given wind conditions, which is usually what concerns the designer.

However, the analysis of a structures response to wind loading may prove to be quite a complex task.

Analysis Using Relatively Short Records:

In order to examine the response of structures to wind loading, research workers have often used the spectral analysis of relatively short lengths of data, but its limitations are still not always realised. In essence, a recording of the response of the structure to the wind is made for a period of several hours. This recording is then split into a number of individual records and an FFT calculated for each one.

The resulting spectra are then averaged together to give an ensemble averaged spectrum from which values of natural frequency, damping and response can be obtained for each mode. This method is usually employed when the response of the structure (e.g., r.m.s. acceleration) to the average wind speed over the recording is required in addition to the dynamic characteristics.

There are three important criteria to consider when using this method:

1. The stationarity of the data,

2. Variance errors, and

3. Bias error.

In order to use spectral analysis methods correctly, it is necessary to have stationary data for analysis. Stationary data have statistical properties which do not vary with time. If the wind direction or speed varies over the length of the recording then the response in any particular mode will vary in amplitude and will not be stationary.

Averaging spectra together which were obtained under different wind conditions will not give the same spectrum that would be obtained from stationary data recorded for the average of these wind conditions, and hence it can be misleading.

Therefore a test for stationarity should be performed on the whole recording. If, as is usually the case, the data are not stationary, then extreme caution should be used in extracting any parameters other than frequency. If damping values are obtained, then they will inevitably overestimate the real values.

Variance or random error results from the fact that any analysis must be performed on a finite number of sample records. On an intuitive level, the more times a random process is sampled, and the results averaged, the greater the accuracy; providing the same process is sampled each time (i.e., it is stationary). Variance error is equal to the reciprocal of the square root of the number of records averaged together.

Therefore, for example, variance error is ±10% for 100 records, but only reduces to ±5% when 400 records are used. Fig. 3 shows the power spectrum for a single record of 1024 seconds duration obtained on a tall building.

Fig. 4 shows the power spectrum obtained by averaging together 100 records from the same building. These 100 records were not obtained consecutively but were chosen by the selective ensemble averaging technique.

Bias errors occur when there is a rapid change of amplitude with respect to frequency and there are too few spectral lines to resolve the variation accurately. Such a situation occurs in the region of a peak in the response spectrum i.e., at a natural frequency. In order to limit bias errors to 2% there must be at least four spectral lines in the half-power bandwidth of each mode.

There are however, two major problems in complying with this. Increasing the number of spectral lines means increasing the frequency resolution. If the number of averages is to be kept the same (in order to keep the variance error the same) then the total time, of the recording must be increased.

This in turn means that the data are less likely to be stationary over the increased length of time. Littler (1992) examined the extreme case of long span suspension bridges. In the case of the Humber Bridge, obtaining 100 records, each with sufficient resolution to give four spectral lines in the half power bandwidth of the fundamental vertical mode, would take approximately 96 hours.

However, in the case of a moderately sized tall building such as the Sheffield Arts Tower the same criteria would require a recording of at least 14 hours. The second problem is that bias error cannot be calculated until the natural frequency and damping of the mode in question are known.

This is a particular problem where forced vibration tests have not been carried out on similar structures. It should be noted that bias errors always lead to damping being overestimated, but it is not possible to calculate the extent, of the overestimation. Also, and most importantly, bias errors will be different for different modes in the same response spectrum. The lower the product of the natural frequency and damping for a particular mode, the larger the bias error.

The half-power bandwidth method of calculating damping from spectra is not ideal when dealing with the results of ambient excitation. This method depends on finding the peak amplitude of response for a particular mode, and then measuring the width of the spectral peak at the half-power amplitude (1/√2 of the peak amplitude).

Bias errors mean that the peak response will always be underestimated, which leads to an underestimation of the half-power amplitude, thus leading to an overestimation of damping. The method of obtaining damping adopted by BRE is to obtain a response spectrum’ from the ambient analysis with normalised displacement as the ordinate.

The best fit theoretical curve to the experimental data is calculated, in a similar fashion to that used in the forced vibration tests. The accuracy of these values can be judged by the fit between the experimental data and the fitted curve.

Use of this method ensures that any damping value obtained is not entirely dependent on the value of the one spectral line closest to the peak response. It is extremely difficult to give a level of accuracy for damping values obtained by this method, and virtually impossible if the data are not stationary. If the data are stationary, then the accuracy is dependent on the inevitable errors in the spectral analysis.

Despite the problems outlined above for obtaining damping values, natural frequencies can be obtained quickly and easily by this method. Satisfactory mode shapes can also be obtained even when non-stationary data are used, providing the response measurements at different positions are made at the same time. This is because the errors in the spectral analysis will be identical for every measurement position. Modal stiffness and therefore modal mass values cannot be obtained by this method as the force level is not known.

Selective Ensemble Averaging:

Selective ensemble averaging tries to overcome the problem of obtaining stationary data of sufficient accuracy by forcing an ensemble to contain stationary data. This is done by making many recordings, or an extremely long continuous recording, and dividing the data into individual blocks, or records, of the length required for analysis, and storing the individual records along with the average wind speed and direction for the length of the record.

Records obtained under similar wind conditions are then selecting and averaged. Fig. 5 shows an example of the data obtained by BRE from Hume Point where this technique was used. Besides the fact that this is likely to be a long and expensive process there are two problems to consider. First in the selecting the ensembles to average, how narrow should the range of wind speeds and direction be in order to obtain accurate results. And second, how can the accuracy of the results be proven.

These problems are interrelated and for the Hume Point experiment the method was proven by producing four ensembles each containing 50 mutually exclusive records but collected under the same wind conditions. The difference in response between the four ensembles was less than that due to variance error, so it was concluded that the limits used to define the wind conditions were sufficiently tight that the data in the whole ensemble could be considered to be stationary.

With enough data, this method can give accurate figures for damping, which can be judged by the goodness of fit in Fig. 5 between the experimental data and the fitted curve. Overall, over 2,250 hours of data were recorded at Hume Point.

However, Fig. 6 from Littler (1992) shows how the technique has been used to assess the damping from the Humber Bridge. The fit is not very good here but it should be noted that this is for a mode with very low frequency and that a total of only 188 hours of data were recorded. Clearly, this technique cannot be used without an accurate way of measuring wind speed and direction at the site.

Selective ensemble averaging is primarily used for obtaining the response of a structure to a given wind to a high degree of accuracy. The results obtained using this method may be used in a comparison with those obtained from calculation or wind tunnel studies to help calibrate codes or wind tunnels.

Although this method yields accurate dynamic characteristics (natural frequency, damping and mode shapes) it is unlikely to be worthwhile employing this technique if it is only these characteristics that are required from the test. The exception to this would be a structure with very low natural frequencies and damping such as a long- span suspension bridge where forced vibration tests or spectral analysis of short ambient recordings is unlikely to be successful.

Other Methods of Analysing Wind Induced Response:

A number of analysis methods have been developed which use much shorter lengths of data than are required by conventional spectral analysis. One of these methods is the autoregressive (AR) model which uses parameters derived by the maximum entropy method (MEM) and this method has been examined using measured data from Hume Point by Cao et al (1995).

Two other related models are the moving average (MA) and the autoregressive moving average (ARMA) which both use parameters derived by the maximum likelihood method. With all of these types of analysis the use of stationary data is very important.

Whilst all of these methods use much shorter lengths of data than standard spectral analysis, the accuracy obtained is dependent upon the model used. If an inappropriate model is selected, then the results will be much worse than those obtained by conventional analysis.

At present, there are no available methods of assessing the likely accuracy of the spectra derived using these methods, and while they may be used for estimating modal frequencies, their use for estimating damping or modal response must be questioned. Nevertheless, the methods have considerable potential, although they may require further refinement and evaluation before they can be used with confidence to evaluate structural characteristics.

Conclusion:

Two major ways of measuring the dynamic characteristics of real structures have been assessed. Steady-state forced vibration testing produces the most accurate assessments, but this will only be the case if both the excitation and the monitoring are carried out with due regard to the possible errors in the testing and analysis procedure.

This method also has the advantage that damping can be assessed by two entirely separate methods and the answers compared. The level of discrepancy between the answers from the two methods gives a good indication of the overall accuracy of the testing procedure.

The spectral analysis of the response of structures to wind loading can also provide useful data and is especially useful for determining the natural frequencies of a structure.

If the data used for the analysis are stationary and both the variance and bias errors involved in the analysis are considered, then information on damping and overall response can be obtained.

The selective ensemble averaging technique avoids the problem of obtaining a sufficiently long continuous record of stationary data, but requires data to be collected for much longer lengths of time. However, it can prove useful for obtaining good quality data for code calibration.