The goal of this project was to perform a modal analysis on a set of data provided by BMW. The data is a result of several impact hammer tests performed on an experimental set-up. Tests were performed with hammer strikes in different places and directions as well as with different numbers of accelerometers placed on the experimental set-up. The modal analysis was treated with a black box approach, taking into account only the signals given by the accelerometers placed inside.
The goal of the modal analysis was to obtain the eigenfrequencies and damping ratios of the system. These values could then formulate a representative rheological modal of the experimental system. The entire analysis was performed using MATLAB.
The following page is derived from a project report written by Giovanni Arcari, Eloisa Baez Jones and Samuel Cummings for the course Modelling and Simulation in Structural Mechanics in the winter term 2019/20.
Experimental Setup
A picture and a detailed schematic representation of the experimental set-up under consideration are shown below:
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| Experimental set-up (Optimol Instruments - detail) |
The set-up consisted of two main parts: an electric motor on the left, directly linked to the part on the right, where a beam-like structure is enclosed in a metallic box. For our project however, we did not consider the details of structural configuration; instead, we focused on the modal testing data and treated the structure itself as a black box.
The three points A, B, C marked in red represent the positions where the impulse-like load was applied to the structure by an impact hammer. The accelerations were then measured by a series of accelerometers ditributed throughout the structure, as shwon in the figure below. This hammer was capable of measuring the applied force so that the response could be normalized. Therefore, the data we received was in units of m/s^2.
For all points, an impulse load was applied in the x-direction. At point B an additional z-direction impulse was applied. The figure below clarifies their position and numbering.
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| Accelerometer setup |
Two more accelerometers, close to points A and C, were attached but are not shown here. Measurements were taken with different numbers of accelerometers in order to determine the effect of their weight on the response of the system. There were four different measurement setups, with 4, 6, 8 and 10 accelerometers. Both sensors at points A and C, numbers 1 and 2 respectively, were always taken into account. For each later measurement, two more were added at a time in the order shown by the numbering in the figure above.
The first measurement is intuitively more representative of the system response, as the mass of the experimental set-up is closest to that of just the structure. On the other hand, accelerometers in different locations makes it possible to detect more details about the response which would otherwise not be detected. These effects were taken into account throughout the data processing by using different data for each assessment; the configuration with the lowest number of sensors was used initially to assess a general shape of the response function, whereas the others were used to detect some details of the response that couldn't be obtained from the previously measured data. The details of the procedure are explained in the following sections.
Measurement Data
This section focuses on the parts of the data that were relevant for this assignment. The experiment consisted of 4 different measurements which were repeated five times. An overview of the measurements collected can be seen in the following table as well the common abbreviation we use to reference a particular data set.
The figure below shows the frequency response for every accelerometer set up and measuring point for the HaXAn impact point. Each subfigure represents one measuring point. For those measuring points which are considered in more than one measurement, more than one line is plotted. The different setups, plotted with different colors, are recalled in the legend. In those graphs, the normalized acceleration (y-axis) is plotted against the frequencies (x-axis), so that it is already possible to grasp an intuitive idea about where the eigenfrequencies could be. The acceleration is normalized by the load, which was measured by the hammer when it impacted the structure.
Influence of Number of Sensors
Before the entire analysis was performed, it was necessary to understand how the presence of the sensors affected the data output. In fact, a severe insertion error would make the entire results meaningless and misleading. It was then appropriate to take a closer look at the different setup curves. In the figure below one can see the effect of mass on an SDOF system with different damping. This highlights that with increasing mass the system amplitude and phase is reduced for all damping cases.
Let us, for example, focus on the response at measuring point 2, due to an impulse load in the x-direction applied at point A, shown in the following figure.
It is assumed here that the applied load, in all four measurements, is exactly the same. The small deviations due to the repetition of the experiment, impossible to quantify, are therefore neglected. Therefore, all the differences in the curves are imputed to the different sensor setups.
In general, two effects are noticeable:
- Decrease in normalized acceleration the first, intuitive effect is that for those setups whose number of sensors is higher, the peaks reach a smaller value. This is due to the fact that, under the same load, the structure with more sensors has a higher mass. The product between mass and acceleration has to remain equal, due to the normalization of the load, so if the masses increases, the acceleration has to be smaller. It must however be kept in mind that this effect doesn't lead to misleading results in terms of eigenfrequencies, whose assessment remains the final goal of the project
- Shift of the peaks: on the other hand, peak shifting more severely affects the eigenfrequency calculation. As it can be seen, increasing the number of sensors shifts the peaks left in the frequency range. As a consequence, it was important for us to carefully select which data to use to assess the eigenfrequencies.
Damping Ratio Assessment
The damping rations were calculated by two different methods, leading to similar results.
Time Series Fitting Approach
The time series approach first requires performing an inverse Fourier transform on the frequency response which was given by the impulse hammer experiment. Then the time series can be fit with a decaying exponential as seen in Eq (1). The coefficient b represents the damping ratio.
| (1) | y = a e^{-b t} |
For our project, we performed an inverse Fourier Transform and used MATLAB's envelope algorithm to fit an envelope around the time domain signal with a single term exponential. The 'peak' algorithm was used, which uses spline interpolation over the local maxima, separated by at least 10 samples in our case. A first guess for the curve fitting algorithm was also generated from the envelope, \vec{E}, using the knowledge that a decaying exponential is what will be fitted to the decaying signal. The equations for this process can be seen in the table in the appendix. The process is illustrated by the upper figure below. A time signal was created for every test and accelerometer set up and a normal distribution was fitted to the data sets, the results of which can be seen in the lower figure below. The mean and standard deviation of the obtained damping ratios for the different test sets are in the table below. Interesting to note from the various normal distributions seen in the lower figure is that there is a nonzero probability that the system could have negative damping, which is not physically possible. This happened because the phase of the accelerometers was not fully taken into account during the testing.
Half Power (-3dB) Method
The half power method uses the bandwidth of a peak at a distance of 3 decibels to estimate the damping ratio. A wider bandwidth corresponds to a larger damping value. After the half power bandwidth {\Delta}f has been obtained, it is inserted into the following equation
| (2) | \xi = \dfrac{{\Delta}f}{2 f_n} |
where f_n is the natural frequency and \xi is the damping ratio. For our project, we first converted the measured acceleration into decibels. Then, the damping ratios were calculated from the peaks corresponding to all the eigenmodes presented in Table on the eigenfrequencies. A calculation of all satisfactory data points yielded an average damping ratio of 0.0471 with a standard deviation of 0.0454.
The table above shows the mean and standard deviation of the damping values obtained from the bandwidth of each eigenfrequency. There were many more samples obtained from the frequencies 290, 372, and 528 Hz, which means that these peaks often had sufficient data to calculate the bandwidth. These peaks had a much higher mean value than those obtained from the frequencies 605, 665, and 702 Hz. Therefore, we took a weighted average respective of the number of samples of each eigenfrequency. This yielded a value of 0.0471, which is closer to the value obtained from the time series method.
In the table below the data is separated according to hammer position. Here the mean values do not vary significantly, and the average is similar to that in the table above. Also, the number of points where a valid damping ratio could be extracted was about the same for each position.
The table above presents the data sorted by measuring point, also referred to as accelerometer position. The measured damping does not change significantly across the apparatus. It is interesting to note that measuring points 3 and 4 had more valid data points than measuring points 1 and 2, even though the latter had an additional round of measurements taken.
The results according to the number of accelerometers present on the apparatus is quite interesting; the damping ratio increased with the number of accelerometers. This increase was most likely caused by the additional mass. It is also possible that the accelerometers themselves oscillated on their glue connections. This would have thrown off the modal behavior of the system, and the glue connections may have also absorbed energy through shear deformation. According to the table, the damping increased by a factor of 1.5 between 4 and 10 accelerometers.
Due to the sparseness of points from the provided data, our Matlab script had to interpolate the points below the peak in order to obtain an accurate bandwidth. We believed a cubic interpolating function would best represent the data. The Matlab function used a shape-preserving cubic interpolation scheme, which may explain why there are some plateaus in the interpolation graphs as seen in Figure (b). The interpolating process is illustrated below, in which the figure below shows the measured response in decibels over the entire frequency range, and Figure (a) shows the results of interpolating the peak at 528 Hz. The red and green markers in the figure below indicate the points from which the bandwidth was calculated. The interpolation was done by inserting points along every 0.5 Hz. The black circles in the figure represent the half power points.
There were many possible sources of error for this method. Firstly, the graph did not decrease by 3 dB on both sides of many peaks, making it much more more difficult and impractical to find the bandwidth. On a related note, some peaks had another peak very close to itself, also rendering bandwidth calculation impractical. Secondly, there were only 90 data points over the entire frequency range. Therefore, some peaks may not have been accurately captured. A probable example of this is shown in Figure (b), where two adjacent points had roughly the same magnitude. This implies that the natural frequency used in Equation (2) would be a value that is on either side of the actual eigenfrequency, and the bandwidth would be larger because the value calculated as 3 dB down would be lower than the actual 3 dB distance. The effects of sparse data points can also be seen in Figure (c), where there are several values that all seem to be a potential peak. This results in a very wide bandwidth. More data would be needed to see the true behavior of the graph here, and to see if it is truly a peak. Lastly, consecutive eigenfrequencies may have an effect on one another when calculating damping with the half power method. Since the properties of our system were not well known and our only data was the acceleration measurements, it would have been very difficult to accurately separate these eigenfrequencies. It is quite interesting to note that all of the reasons listed above except for the contribution of the modes on one another would cause the damping ratio to be larger. All of these points most likely contributed to inaccuracies in the reported value for the damping ratio.
Comparison of Damping Calculation Methods
The primary difference between the two methods used here is that the half power method calculates the damping in the frequency domain, whereas the time series approach uses the time domain. Although both methods used data from each accelerometer in each hammer position, the half power method was not able to extract the bandwidth from every eigenfrequency due to insufficient peak depths. However, this method allowed us to compare damping ratios through many different perspectives, as shown in the tables in the previous section. The time dependent method also calculates the damping independently of the individual eigenfrequencies, which may have lead to a more accurate overall result.
The average damping ratio obtained from the time series approach was 0.0567, and the average from the half power method was 0.0471. When working with data from a real system in contrast to an idealized system, the time series approach is most likely more reliable. As discussed above, the half power method has many drawbacks that can all have an effect if there are fluctuations in the data. The time series approach on the other hand uses a fitted decaying exponential curve, which will be more resilient to data fluctuations. Therefore, we would assert that the time series approach is better for an overall damping ratio value, but the half power method is useful for viewing the effects of different aspects on the damping.
Eigenfrequencies
Peak Finding for Eigenfrequency Assessment
In order to estimate the eigenfrequencies, an analysis to find the peaks of each response function was performed. This means that each measurement, measuring point and load case were considered at a time. Then, the Matlab built in function 'findpeaks' was applied. According to the literature, findpeaks(Y) finds local peaks in the data vector \textbf{Y}. A local peak is defined as a data sample which is either larger than the two neighboring samples or is equal to Inf.
By default the function identifies a peak as any point whose image is greater than the one of the points directly on the left and on the right. Therefore, the result of the function application looks like the image shown in the following, where the peaks are highlighted by small red circles.
As can be seen from the picture, this yielded misleading results. In fact, whereas on the left the situation was as desired, in the low frequency range it is clear that some fake peaks were detected, due to some noise in the measuring data. In order to get rid of this effect, a threshold was set, such that only the desired peaks were considered. After this, one of the results looked like this:
This result was considered satisfactory: all the major peaks were in fact detected, and no problem due to experimental noise occurred. From this point on, the eigenfrequency analysis was based on the data calculated from this function.
Peak Occurrences Assessment
In order to collect the data from all the measurements, and to assess an estimate for the eigenfrequencies, a histogram was created by counting the number of occurrences of peaks in 10 Hz ranges. The data were collected in 4 separate graphs, one for each sensor setup. As previously mentioned, the setup with only 4 sensors was used wherever possible so that the insertion error would be minimized. The histogram for the setup with the lowest number of sensors is shown in the following figure.
If bands with small occurrence were neglected, it was possible to find 5 possible eigenfrequencies by this graph. Here, not only the occurrence of a single band was taken into consideration; in some cases such as for the two bands in the 280-300 Hz range, the sum of the two values was taken as relevant in order to account for the possible split-up due to the discretization of the frequency range. In the following table, the 5 detected eigenfrequencies are shown.
Modal Fit
In order to compare the values obtained by our eigenfrequency method, the MATLAB built in function called 'modalfit', which estimates the eigenfrequencies of a system, was also implemented.
The MATLAB function takes the input of one or multiple frequency response functions, sampling frequency, and the number of modes of the system. The function then estimates the impulse response function (meaning the system response in the time domain). Then it fits to each impulse response a series of complex damped sinusoids (Eq. (3)) using Prony's method where the parameters of the fit correspond to the modal parameters of the system.
| (3) | s_{i}(n) = A_{i} \, e^{-b_{i}\,n\,/f_{s}} \, cos\left(2 \pi f_{i} n/f_{s} + \phi_{i} \right) |
f_{i} are the frequency components from which the eigenvalues can be calculated from. f_{s} is the sampling frequency. b_{i} is the damping coefficient. Following this method, the obtained eigenfrequencies and damping are seen in the table below. The eigenfrequencies of the eight measuring points are given as the modalfit was used eight times, each time on the set of frequency responses for each measuring point. For example: for the first measuring point the frequency responses for every hammer impact location and every accelerometer setup was given. This can be seen for the first measuring point can be seen in the figure below. The lines denoted by modalfit are reconstructed frequency responses which modalfit formulates based on the strength of the resonance, the estimated eigenvalues, and damping from Prony's method. Eigenfrequencies listed as '-' in the table below are frequencies which were out of the frequency range given to modalfit which it does not return. As modalfit also takes the number of modes as input different number of modes were tested as seen in the figure below.
Appendix
Amplitude and Phase Response of Force Excited Damped SDOF
Eigenfrequencies and damping obtained from modalfit with 3 modes
Reconstructed Frequency Responses for All Tests at First Measuring Point
Equations Required for the Finding Initial Guess of Exponential Function \textbf{C} is of the size [\text{signallength } \cdot 2]
References
Integrated Systems Research Inc. Estimating Damping Values Using the Half Power Method . https://www.isrtechnical.com/media/tech-briefs/estimating_damping_values.pdf
Tom Irvine. The Half Power Bandwidth Method for Damping Calculation . http://www.vibrationdata.com/tutorials2/half_power_bandwidth.pdf
MATLAB. https://www.mathworks.com/help/signal/ref/modalfit.html
Brown University. https://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations_forced.htm