Computational Exercises
Using multiple absorbers
In the vibration absorber short lab you have looked at the effect of adding a tuned mass damper (TMD) to the structure to reduce the amplitude of vibration at resonance. While this was accomplished, the TMD also introduced two additional resonant peaks, at frequencies each side of the original peak. In an earthquake, excitation of the structure occurs over a range of frequencies, and these two peaks (even though lower) may still produce unacceptably large vibration. Additionally, in a real structure, it may be more practical to have multiple small absorbers, rather than a few large ones. In this experiment, you are to investigate the use of multiple absorbers, tuned such that they spread the peaks and reduce vibration over a range of frequencies.
Using an Octave/Matlab model of the structure, carry out a similar analysis to that done in the vibration absorber short lab, but using multiple absorbers. Here are a few suggested questions to investigate:
- Compare, for example, the effects of using 1, 10, 100 and 1000 absorbers tuned to slightly different frequencies (with the same total mass in each case). Which gives the widest spread?
- What is the minimum total mass of absorbers required to reduce the maximum (harmonic) response of the structure to, say, 10% of its value with no absorbers?
- How does this change if you analyse the impulse response? Is there are trade-off between the optimum tuning to deal with harmonic and impulse responses?
- Can you think of a way to test these predictions on the structure?
This extended exercise is based around the simulation of the structure, so you will need to understand Octave/Matlab code such as that used in the signal processing short lab.
M-file code for analysing the system
Designing a full-size absorber to cope with a real earthquake
In this extended exercise, you are required to produce initial design parameters for a vibration absorber on a full-scale structure to respond to earthquakes. There are a number of suggested items here; you are not necessarily expected to answer all of the questions in detail, but to develop your own plan about what to investigate and how:
- Test the vibration absorber design from the short lab on the model structure. How well do your predictions compare with the measured results? Can you tune it precisely to a resonant frequency?
- Perform a Fourier analysis on ground movement data from an earthquake (data files are supplied - links below) to find the major frequency components.
- Use dimensional analysis either to scale up the parameters of the model to a full-size structure, or to modify the input data so it is suitable for use with the model.
- How do the frequency components present in the earthquake data compare to the resonant frequencies of the model structure? What is the most structurally onerous condition?
- Take these results and design the parameters of a vibration absorber suitable for use on a full-size building.
- Do your results agree in principal with absorbers used on full-size structures? How might you test the performance of your absorber in response to realistic earthquake loading?
Further analysis of sampled signals and their responses
A number of signal processing concepts were introduced in the Fourier Analysis short lab. This extended exercise gives you the opportunity to examine some of these effects in more detail.
- Note that in the short lab each sample was run twice and only the results from the second run were recorded. Try altering the code to only run the sample once and compare the two results. Explain the differences.
- For a given signal, investigate the minimum sampling time you must use (even running it twice) to obtain good results.
- If you sample for 10 seconds with an input signal of 5Hz, the section of signal that you take will contain a whole number of periods. Look at the response for such a sample and then investigate what happens if this is not the case (e.g. sample a 5Hz signal for 10.1 seconds). Explain the differences in these two cases.
Additionally you may want to investigate more detailed aspects of the system response. You may have noticed that when using the sweep function there is considerable noise above the highest generated frequency. This is because both channels have very small signals in this frequency range (look at the transforms to confirm this). One way to reduce or eliminate this is to use a sweep that also covers these frequencies, but then the sample time must be increased to get acceptable results in the region of interest (i.e. at lower frequencies where the natural frequencies of the structure occur). How can you solve this problem? A hint is that the sweep function, as used, is linear, so you might want to think of other possibilities. Investigate, for example, the Matlab inbuilt function 'chirp'. in which the frequency varies exponentially over time. (Look closely at the start and end frequencies of the chirp and explain why they are not what you might expect).
Windowing
The DFT algorithm, like the Fourier transform, is designed to operate on a continuous signal (i.e., continuous in time). As it is not possible to measure a signal for all time, the continuous signal is synthesised from that measured during a finite time period (period T). This will usually produce a discontinuity in the signal when an attempt is made to match up the first and last sampled points.
This causes a problem, as the Fourier transform treats the discontinuity as part of the original function, leading to spurious results (termed spectral leakage). This can be mitigated by the use of windowing. A window functions multiplies the time domain signal to reduce or remove this discontinuity. This is demonstrated in the diagrams below using a Hamming window - this is a particular type of window function, of which there are many.
While the use of a window function can be beneficial, it can also have drawbacks. In this extended exercise, you are asked to investigate these effects further.
© Cambridge University, Engineering Department
Last updated 12/10/2012 by hrs@eng