Group Assignment <2.3>: Analyzing cantilever-beam accelerations and global Mean Sea-Level measurements

Group Assignment <2.3>: Analyzing cantilever-beam accelerations and global Mean Sea-Level measurements#

Every task 1 through 9 is worth 1 point; a score of 9 yields a grade of 10 for this group assignment. Some of the questions in the notebook and report are very similar. Only the answer in your report will be graded.

Task 1#

Create a time-array starting at $t=0$ s, ending at $t=5.0$ s, with a sampling rate of 100 Hz. Please note down how many samples there are and what the last value should be.

Hint: what is the number of samples $N$? And what should then be the last value in the time-array? Note that, for example, with $N=10$ samples at a sampling rate of $f_s = 100$ Hz, we have signal samples at times $t=0.00, 0.01, 0.02, ... , 0.08, 0.09$ seconds, hence $T_{meas}= N \Delta t =0.1$ seconds (the sample-and-hold convention).

Make a plot of the signal against time. Note that this is strictly a sampled signal $x_n$ rather than $x(t)$, but since we use a rather high sampling rate, the signal shown is close to continuous in time. Connecting the sample points of $x_n$ in the graph by lines, as done in the graph below, corroborates the suggestion of a continuous-time signal. Be aware!

The number of samples is $N=\frac{T_{meas}}{\Delta t}=T_{meas}\cdot f_s=5\cdot 100=500$. The last value should be $4.99$ because we use a $\Delta t=\frac{1}{f_s}=0.01$ s.

Task 2#

Include the plot of the modulus of the Fourier transform against frequency $f$ in your report. On top of that, answer to the following questions (include quantitative values):

Describe the amplitude spectrum (how does it look like).
Do you notice anything peculiar about the amplitude spectrum (something strange)?
A peak of 0.5 should be found (from theory) at a frequency of 1 Hz and one around 100 Hz (exactly at 99 Hz), the rest all close to zero.
The peak at 100 Hz (or 99 Hz to be precise) should not be there; that’s strange. We did not input a signal with such a frequency.

Task 3#

Repeat Task 2 with different measurement times $T_{meas}$ for the signal. Use measurement times such that the $f_c$ = 1 Hz oscillation fits exactly 1 time, 5 times and 20 times.
Plot the amplitude spectrum for all three measurement times, only for positive frequencies, in separate graphs (log-log scale) with the same domains (include the three plots in your report), and answer to the following questions:
What is the effect of changing $T_{meas}$ on the frequency range in the amplitude spectrum? Does the highest analysis frequency change?
Does the frequency resolution change?

Include plots to support your arguments. Numerical analysis beyond interpreting the graphs is not required.

No, the highest analysis frequency stays the same (as it is related to the sampling frequency, which we did not change).
Yes, the frequency resolution becomes better/finer (frequence steps gets smaller).

Task 4#

Include the two plots in your report. Then answer to the following questions (please include quantitative values and variable names where relevant.):

What do you see in the frequency plot? Are there peaks? How many? Where?
Does this match what you see in the time plot?
Will changing the measurement time (duration) help to solve this mis-match?
There are two peaks, one of $0.5$ at $1$ Hz and one of $0.05$ at $20$ Hz, which should be at $80$ Hz (again exact values are hard to read but should be derived from theory).
Yes, there is a large amplitude sinusoid with a frequency of $1$ Hz, and a small amplitude sinusoid with a frequency of $20$ Hz on top if it (count the number of wiggles - there are $20$ wavies within 1 second). However, it does not match the input signal we created (with $1$ Hz and $80$ Hz). Mind: though the time plot may suggest continuous-time, we’re working here with discrete-time signals.
No, the frequency resolution would change, but not the largest frequency of the amplitude spectrum, as that one is determined by the sampling frequency $f_s$.

Task 5#

Include the 4 times two graphs in your report. Then answer the following questions:

At what frequency does the (aliased) 80 Hz signal appear in the spectrum, for the above values of $f_s$ (provide numerical answers)?
Can you figure out the relationship (a simple equation) between the sample rate and the frequency of the original signal, and the frequency at which the alias appears in the spectrum plot?

Include numerical values in both your answers.

Different frequencies:
- $f_s=110$ Hz: peak at $30$ Hz.
- $f_s=150$ Hz: peak at $70$ Hz.
- $f_s=160$ Hz: no peak (the $80$ Hz signal is sampled exactly twice per cycle, exactly at the ‘zero-passes’; we now don’t see it at all).
- $f_s=200$ Hz: peak at $80$ Hz.
The sample rate needs to be more than twice the (highest) frequency of the signal, which is the Nyquist rate. As long as we do not meet this requirement the alias appears mirrored about the Nyquist frequency (which is half the sampling frequency), e.g. with $f_s=110$ Hz, the alias of $f_i=80$ Hz appears at $110-80=30$ Hz (it is the $f=f_i=-80$ Hz at the first copy $k=1$ of the spectrum $X(f)$, $\sum_{k=-\infty}^{\infty} X(f-k f_s)$).

Task 6#

Include the two plots in your report. Then answer the following questions:

Do you see any changes in the time plot, compared to the earlier plot? Describe them!
What is the dominant frequency of the signal now? What does the amplitude spectrum look like?
The signal is now clearly a sinusoid with an exponentially decaying amplitude, a.k.a. a damped vibration!
Around $5$ Hz (max value of $|X_{cont}|$ at $4.98$ Hz).

Instead of a ‘crisp’ spectrum, with a very clear, distinct and ultimately narrow peak, you now get a kind of smoothed or faded peak. This is simply the result of dealing with a damped harmonic, rather than a perfect harmonic. The signal somehow still looks pretty periodic, but strictly spoken, the signal is not periodic anymore, as the amplitude slightly changes (decreases) with time.

Task 7#

Include the plot of the input signal as a function of time in your report, and note down the estimated offset and slope of the trend (i.e. numerical values).

The mean is $-0.43$ m/s², so that’s considerable, and the slope is $-1.3e^{-4}$ m/s², which is pretty much negligible over the duration of this experiment; mean/offset when one forgets to subtract $t[0]$ from the time column in the A-matrix, hence reporting the offset for 00:00h UTC, this is $8.08$ m/s².

Task 8#

Compute and plot the periodogram for the detrended accelerometer measurements of Task 7 (if you prefer, feel free to use a linear scaling of the axes here, rather a log-log, and, use $T$ as defined already in the code of Task 7). Please, pay attention to correctly labelling the axes, and stating units of the quantities along the axes! Include the resulting periodogram plot in your report.

Task 9#

considering the plot of Task 8, answer the following questions:

Report the damped natural frequency (in Hertz) of this one-degree-of-freedom (1DOF) mechanical system. Does it match the motion of the beam shown in the cantilever-beam video?
The acceleration was measured at quite a high sample rate of $50$ Hz. What is the minimum sampling frequency to correctly identify the damped natural frequency in the periodogram?

Around $1.36$ Hz, and yes, this matches the motion in the video.
Nyquist rate is twice the (highest) frequency in the signal, hence $2 \times 1.36 = 2.72$ Hz, so, any value larger than $2.72$ Hz (or slightly larger actually, as it is a damped harmonic; the peak is a bit wider, like in Task 6).

Task 10 (optional, not for grade)#

By Christiaan Tiberius and Caspar Jungbacker, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.