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!

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)?

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.

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?

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.

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?

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).

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:

  1. 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?
  2. 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?

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.

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