diff --git a/modal-analysis/data/frf_processing.zip b/modal-analysis/data/frf_processing.zip index 36a91e6..73a79a1 100644 Binary files a/modal-analysis/data/frf_processing.zip and b/modal-analysis/data/frf_processing.zip differ diff --git a/modal-analysis/data/modal_frf_coh.zip b/modal-analysis/data/modal_frf_coh.zip index cc408e0..e5e73f4 100644 Binary files a/modal-analysis/data/modal_frf_coh.zip and b/modal-analysis/data/modal_frf_coh.zip differ diff --git a/modal-analysis/figs/compare_acc_x_dir.png b/modal-analysis/figs/compare_acc_x_dir.png index 9ec3cf4..1d24023 100644 Binary files a/modal-analysis/figs/compare_acc_x_dir.png and b/modal-analysis/figs/compare_acc_x_dir.png differ diff --git a/modal-analysis/figs/compare_acc_y_dir.png b/modal-analysis/figs/compare_acc_y_dir.png index 77a36e8..94003bf 100644 Binary files a/modal-analysis/figs/compare_acc_y_dir.png and b/modal-analysis/figs/compare_acc_y_dir.png differ diff --git a/modal-analysis/figs/frf_all_bodies_one_direction.png b/modal-analysis/figs/frf_all_bodies_one_direction.png index d7bd783..86d02d5 100644 Binary files a/modal-analysis/figs/frf_all_bodies_one_direction.png and b/modal-analysis/figs/frf_all_bodies_one_direction.png differ diff --git a/modal-analysis/figs/recovered_frf_comparison_hexa.png b/modal-analysis/figs/recovered_frf_comparison_hexa.png index 4f9678f..007a534 100644 Binary files a/modal-analysis/figs/recovered_frf_comparison_hexa.png and b/modal-analysis/figs/recovered_frf_comparison_hexa.png differ diff --git a/modal-analysis/figs/recovered_frf_comparison_ty.png b/modal-analysis/figs/recovered_frf_comparison_ty.png index 5a1c9f2..ac69ef2 100644 Binary files a/modal-analysis/figs/recovered_frf_comparison_ty.png and b/modal-analysis/figs/recovered_frf_comparison_ty.png differ diff --git a/modal-analysis/figs/relative_motion_comparison.png b/modal-analysis/figs/relative_motion_comparison.png index ad90d72..7d96d19 100644 Binary files a/modal-analysis/figs/relative_motion_comparison.png and b/modal-analysis/figs/relative_motion_comparison.png differ diff --git a/modal-analysis/frf_processing.html b/modal-analysis/frf_processing.html index 7959ff2..a07e6b3 100644 --- a/modal-analysis/frf_processing.html +++ b/modal-analysis/frf_processing.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Modal Analysis - Processing of FRF @@ -280,14 +280,14 @@ for the JavaScript code in this tag.

Table of Contents

@@ -328,8 +328,8 @@ All the files (data and Matlab scripts) are accessible -

1 Importation of measured FRF curves

+
+

1 Importation of measured FRF curves

We load the measured FRF and Coherence matrices. @@ -344,11 +344,11 @@ load( -

2 From accelerometer DOFs to solid body DOFs - Mathematics

+
+

2 From accelerometer DOFs to solid body DOFs - Mathematics

-Let's consider the schematic shown on figure 1 where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z. +Let's consider the schematic shown on figure 1 where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.

@@ -356,14 +356,14 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of

-
+

local_to_global_coordinates.png

Figure 1: Schematic of the measured motions of a solid body

-From the figure 1, we can write: +From the figure 1, we can write:

\begin{align*} \vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\ @@ -432,8 +432,8 @@ This inversion is equivalent to resolving a mean square problem.
-
-

3 What reference frame to choose?

+
+

3 What reference frame to choose?

The question we wish here to answer is how to choose the reference frame \(\{O\}\) in which the DOFs of the solid bodies are defined. @@ -453,7 +453,7 @@ The possibles choices are:

  • Base located at the joint position: this is where we want to see the motion and estimate stiffness
    • - +
    @@ -497,8 +497,8 @@ As the easiest choice is to choose a common frame, we start with that solution. -
    -

    4 From accelerometer DOFs to solid body DOFs - Matlab Implementation

    +
    +

    4 From accelerometer DOFs to solid body DOFs - Matlab Implementation

    First, we initialize a new FRF matrix FRFs_O which is an \(n \times p \times q\) with: @@ -563,26 +563,26 @@ Then, as we know the positions of the accelerometers on each solid body, and we

    -
    -

    5 Analysis of some FRF in the global coordinates

    +
    +

    5 Analysis of some FRF in the global coordinates

    -First, we can compare the motions of the 6 solid bodies in one direction (figure 2) +First, we can compare the motions of the 6 solid bodies in one direction (figure 2)

    -We can also compare all the DOFs of one solid body (figure 3). +We can also compare all the DOFs of one solid body (figure 3).

    -
    +

    frf_all_bodies_one_direction.png

    Figure 2: FRFs of all the 6 solid bodies in one direction

    -
    +

    frf_one_body_all_directions.png

    Figure 3: FRFs of one solid body in all its DOFs

    @@ -590,8 +590,8 @@ We can also compare all the DOFs of one solid body (figure
    -
    -

    6 Comparison of the relative motion of solid bodies

    +
    +

    6 Comparison of the relative motion of solid bodies

    Now that the motion of all the solid bodies are expressed in the same frame, we should be able to compare them. @@ -609,11 +609,11 @@ Then, if \(\Delta_{ij,x} \ll 0\) in the frequency band of interest, we have that

    -This normalized relative motion is shown on figure 4 for all the directions and for all the adjacent pair of solid bodies. +This normalized relative motion is shown on figure 4 for all the directions and for all the adjacent pair of solid bodies.

    -
    +

    relative_motion_comparison.png

    Figure 4: Relative motion between each stage

    @@ -629,8 +629,8 @@ The relative motion of two solid bodies may be negligible when exciting the stru
    -
    -

    7 Verify that we find the original FRF from the FRF in the global coordinates

    +
    +

    7 Verify that we find the original FRF from the FRF in the global coordinates

    We have computed the Frequency Response Functions Matrix FRFs_O representing the response of the 6 solid bodies in their 6 DOFs. @@ -664,7 +664,6 @@ This will help us to determine if: % We get the position of the accelerometer expressed in frame O pos = acc_pos(acc_i, :)'; posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0]; - [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0] FRF_recovered(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0; end @@ -678,24 +677,24 @@ We then compare the original FRF measured for each accelerometer with the recove

    -The FRF for the 4 accelerometers on the Hexapod are compared on figure 5. +The FRF for the 4 accelerometers on the Hexapod are compared on figure 5. All the FRF are matching very well in all the frequency range displayed.

    -The FRF for accelerometers located on the translation stage are compared on figure 6. +The FRF for accelerometers located on the translation stage are compared on figure 6. The FRF are matching well until 100Hz.

    -
    +

    recovered_frf_comparison_hexa.png

    Figure 5: Comparison of the original FRF with the recovered ones - Hexapod

    -
    +

    recovered_frf_comparison_ty.png

    Figure 6: Comparison of the original FRF with the recovered ones - Ty

    @@ -716,8 +715,8 @@ This valid the fact that a multi-body model can be used to represent the dynamic
    -
    -

    8 Saving of the FRF expressed in the global coordinates

    +
    +

    8 Saving of the FRF expressed in the global coordinates

    save('mat/frf_o.mat', 'FRFs_O');
@@ -728,7 +727,7 @@ This valid the fact that a multi-body model can be used to represent the dynamic

    Author: Dehaeze Thomas

    -

    Created: 2019-07-05 ven. 11:06

    +

    Created: 2019-07-05 ven. 11:49

    Validate

    diff --git a/modal-analysis/frf_processing.org b/modal-analysis/frf_processing.org index cf98679..354b472 100644 --- a/modal-analysis/frf_processing.org +++ b/modal-analysis/frf_processing.org @@ -276,7 +276,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al figure; - ax1 = subaxis(2, 1, 1); + ax1 = subplot(2, 1, 1); hold on; for solid_i = solids_i plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i}); @@ -288,7 +288,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al legend('Location', 'northwest'); title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i})); - ax2 = subaxis(2, 1, 2); + ax2 = subplot(2, 1, 2); hold on; for solid_i = solids_i plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180); @@ -375,7 +375,7 @@ This normalized relative motion is shown on figure [[fig:relative_motion_compari figure; for i = 2:6 - subaxis(3, 2, i); + subplot(3, 2, i); hold on; for dir_i = dirs_i H = (squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :))-squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))./(abs(squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :)))+abs(squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))); @@ -473,7 +473,7 @@ The FRF are matching well until 100Hz. for i = 1:length(accs_i) acc_i = accs_i(i); - subaxis(2, 2, i); + subplot(2, 2, i); hold on; for dir_i = 1:3 @@ -525,7 +525,7 @@ The FRF are matching well until 100Hz. for i = 1:length(accs_i) acc_i = accs_i(i); - subaxis(2, 2, i); + subplot(2, 2, i); hold on; for dir_i = 1:3 diff --git a/modal-analysis/mat/frf_coh_matrices.mat b/modal-analysis/mat/frf_coh_matrices.mat index 7175cfd..9853574 100644 Binary files a/modal-analysis/mat/frf_coh_matrices.mat and b/modal-analysis/mat/frf_coh_matrices.mat differ diff --git a/modal-analysis/matlab/frf_processing.m b/modal-analysis/matlab/frf_processing.m index dcc13af..97be3b7 100644 --- a/modal-analysis/matlab/frf_processing.m +++ b/modal-analysis/matlab/frf_processing.m @@ -75,7 +75,7 @@ exc_dir = 1; figure; -ax1 = subaxis(2, 1, 1); +ax1 = subplot(2, 1, 1); hold on; for solid_i = solids_i plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i}); @@ -87,7 +87,7 @@ ylabel('Amplitude'); legend('Location', 'northwest'); title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i})); -ax2 = subaxis(2, 1, 2); +ax2 = subplot(2, 1, 2); hold on; for solid_i = solids_i plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180); @@ -160,7 +160,7 @@ exc_dir = 1; figure; for i = 2:6 - subaxis(3, 2, i); + subplot(3, 2, i); hold on; for dir_i = dirs_i H = (squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :))-squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))./(abs(squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :)))+abs(squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))); @@ -244,7 +244,7 @@ figure; for i = 1:length(accs_i) acc_i = accs_i(i); - subaxis(2, 2, i); + subplot(2, 2, i); hold on; for dir_i = 1:3 @@ -292,7 +292,7 @@ figure; for i = 1:length(accs_i) acc_i = accs_i(i); - subaxis(2, 2, i); + subplot(2, 2, i); hold on; for dir_i = 1:3 diff --git a/modal-analysis/matlab/modal_frf_coh.m b/modal-analysis/matlab/modal_frf_coh.m index d30bf9f..442ba3c 100644 --- a/modal-analysis/matlab/modal_frf_coh.m +++ b/modal-analysis/matlab/modal_frf_coh.m @@ -417,7 +417,7 @@ acc_i = [1 , 4 ; figure; for i = 1:size(acc_i, 1) - subaxis(3, 3, i); + subplot(3, 3, i); hold on; plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) @@ -459,7 +459,7 @@ acc_i = [1, 2; figure; for i = 1:size(acc_i, 1) - subaxis(3, 3, i); + subplot(3, 3, i); hold on; plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) diff --git a/modal-analysis/measurement.html b/modal-analysis/measurement.html index b6d5ad2..061b091 100644 --- a/modal-analysis/measurement.html +++ b/modal-analysis/measurement.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Modal Analysis - Measurement @@ -280,34 +280,34 @@ for the JavaScript code in this tag.

    Table of Contents

    @@ -319,8 +319,8 @@ All the files (data and Matlab scripts) are accessible -

    1 Goal

    +
    +

    1 Goal

    The goal is to measure the dynamic of the Micro-Station and to extract Frequency Response Functions. @@ -328,20 +328,20 @@ The goal is to measure the dynamic of the Micro-Station and to extract Frequency

    -
    -

    2 Instrumentation Used

    +
    +

    2 Instrumentation Used

    In order to perform to Modal Analysis and to obtain first a Response Model, the following devices are used:

      -
    • An acquisition system (OROS) with 24bits ADCs (figure 1)
      • -
    • 3 tri-axis Accelerometers (figure 2) with parameters shown on table 1
      • -
    • An Instrumented Hammer with various Tips (figure 3) (figure 4)
      • +
    • An acquisition system (OROS) with 24bits ADCs (figure 1)
      • +
    • 3 tri-axis Accelerometers (figure 2) with parameters shown on table 1
      • +
    • An Instrumented Hammer with various Tips (figure 3) (figure 4)
    -
    +

    oros.png

    Figure 1: Acquisition system: OROS

    @@ -354,13 +354,13 @@ Anti-aliasing filters are also included in the system.

    -
    +

    accelero_M393B05.png

    Figure 2: Accelerometer used: M393B05

    -
    Table 1: Advantages and disadvantages for the choice of reference frame
    +
    @@ -403,14 +403,14 @@ It excites more the low frequency range where the coherence is low, the overall

    -
    +

    instrumented_hammer.png

    Figure 3: Instrumented Hammer

    -
    +

    hammer_tips.png

    Figure 4: Hammer tips

    @@ -422,12 +422,12 @@ The accelerometers are glued on the structure.
    -
    -

    3 Structure Preparation and Test Planning

    +
    +

    3 Structure Preparation and Test Planning

    -
    -

    3.1 Structure Preparation

    +
    +

    3.1 Structure Preparation

    All the stages are turned ON. @@ -470,8 +470,8 @@ All other elements have been remove from the granite such as another heavy posit

    -
    -

    3.2 Test Planing

    +
    +

    3.2 Test Planing

    The goal is to identify the full \(N \times N\) FRF matrix (where \(N\) is the number of degree of freedom of the system). @@ -504,8 +504,8 @@ The measurement thus consists of:

    -
    -

    3.3 Location of the Accelerometers

    +
    +

    3.3 Location of the Accelerometers

    4 tri-axis accelerometers are used for each solid body. @@ -524,11 +524,11 @@ The position of the accelerometers are:

    • 4 on the first granite
      • -
    • 4 on the second granite (figure 5)
      • -
    • 4 on top of the translation stage (figure 6)
      • +
    • 4 on the second granite (figure 5)
      • +
    • 4 on top of the translation stage (figure 6)
    • 4 on top of the tilt stage
    • 3 on top of the spindle
      • -
    • 4 on top of the hexapod (figure 7)
      • +
    • 4 on top of the hexapod (figure 7)

    @@ -536,43 +536,43 @@ In total, 23 accelerometers are used: 69 DOFs are thus measured.

    -The position and orientation of all the accelerometers used are shown on figure 8. +The position and orientation of all the accelerometers used are shown on figure 8.

    -The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure 9). +The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure 9).

    -
    +

    accelerometers_granite2_overview.jpg

    Figure 5: Accelerometers located on the top granite

    -
    +

    accelerometers_ty_overview.jpg

    Figure 6: Accelerometers located on top of the translation stage

    -
    +

    accelerometers_hexa_overview.jpg

    Figure 7: Accelerometers located on the Hexapod

    -
    +

    nass-modal-test.png

    Figure 8: Position and orientation of the accelerometer used

    -
    +

    location_accelerometers.png

    Figure 9: Position of the accelerometers using SolidWorks

    @@ -599,9 +599,9 @@ acc_pos = acc_pos(

    -The positions of the sensors relative to the point of interest are shown below (table 2). +The positions of the sensors relative to the point of interest are shown below (table 2).

    -
    Table 1: 393B05 Accelerometer Data Sheet
    +
    @@ -787,15 +787,15 @@ The positions of the sensors relative to the point of interest are shown below ( -
    -

    3.4 Hammer Impacts

    +
    +

    3.4 Hammer Impacts

    Only 3 impact points are used.

    -The impact points are shown on figures 10, 11 and 12. +The impact points are shown on figures 10, 11 and 12.

    @@ -803,21 +803,21 @@ We chose this excitation point as it seems to excite all the modes in the freque

    -
    +

    hammer_x.gif

    Figure 10: Hammer Blow in the X direction

    -
    +

    hammer_y.gif

    Figure 11: Hammer Blow in the Y direction

    -
    +

    hammer_z.gif

    Figure 12: Hammer Blow in the Z direction

    @@ -826,19 +826,19 @@ We chose this excitation point as it seems to excite all the modes in the freque
    -
    -

    4 Signal Processing

    +
    +

    4 Signal Processing

    -
    -

    4.1 Averaging

    +
    +

    4.1 Averaging

    The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise. -The parameters for the impact test are shown on table 3. +The parameters for the impact test are shown on table 3.

    -
    Table 2: position of the accelerometers
    +
    @@ -876,15 +876,15 @@ The parameters for the impact test are shown on table 3 -
    -

    4.2 Windowing

    +
    +

    4.2 Windowing

    Windowing is used on the force and response signals.

    -A boxcar window (figure 13) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless. +A boxcar window (figure 13) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless. The parameters are:

      @@ -893,14 +893,14 @@ The parameters are:
    -
    +

    windowing_force_signal.png

    Figure 13: Window used for the force signal

    -An exponential window (figure 14) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal. +An exponential window (figure 14) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal. The parameters are:

      @@ -917,7 +917,7 @@ The parameters are:
    -
    +

    windowing_response_signal.png

    Figure 14: Window used for the response signals

    @@ -926,8 +926,8 @@ The parameters are:
    -
    -

    5 Force and Response signals

    +
    +

    5 Force and Response signals

    Let's load some obtained data to look at the force and response signals. @@ -939,33 +939,33 @@ Let's load some obtained data to look at the force and response signals.

    -
    -

    5.1 Raw Force Data

    +
    +

    5.1 Raw Force Data

    -The force input for the first measurement is shown on figure 15. We can see 10 impacts, one zoom on one impact is shown on figure 16. +The force input for the first measurement is shown on figure 15. We can see 10 impacts, one zoom on one impact is shown on figure 16.

    -The Fourier transform of the force is shown on figure 17. This has been obtained without any windowing. +The Fourier transform of the force is shown on figure 17. This has been obtained without any windowing.

    -
    +

    raw_data_force.png

    Figure 15: Raw Force Data from Hammer Blow

    -
    +

    raw_data_force_zoom.png

    Figure 16: Raw Force Data from Hammer Blow - Zoom

    -
    +

    fourier_transfor_force_impact.png

    Figure 17: Fourier Transform of the 10 force impacts for the first measurement

    @@ -973,33 +973,33 @@ The Fourier transform of the force is shown on figure 17
    -
    -

    5.2 Raw Response Data

    +
    +

    5.2 Raw Response Data

    -The response signal for the first measurement is shown on figure 18. One zoom on one response is shown on figure 19. +The response signal for the first measurement is shown on figure 18. One zoom on one response is shown on figure 19.

    -The Fourier transform of the response signals is shown on figure 20. This has been obtained without any windowing. +The Fourier transform of the response signals is shown on figure 20. This has been obtained without any windowing.

    -
    +

    raw_data_acceleration.png

    Figure 18: Raw Acceleration Data from Accelerometer

    -
    +

    raw_data_acceleration_zoom.png

    Figure 19: Raw Acceleration Data from Accelerometer - Zoom

    -
    +

    fourier_transform_response_signals.png

    Figure 20: Fourier transform of the measured response signals

    @@ -1007,15 +1007,15 @@ The Fourier transform of the response signals is shown on figure -

    5.3 Computation of one Frequency Response Function

    +
    +

    5.3 Computation of one Frequency Response Function

    Now that we have obtained the Fourier transform of both the force input and the response signal, we can compute the Frequency Response Function from the force to the acceleration.

    -We then compare the result obtained with the FRF computed by the modal software (figure 21). +We then compare the result obtained with the FRF computed by the modal software (figure 21).

    @@ -1032,7 +1032,7 @@ In the following analysis, FRF computed from the software will be used.

    -
    +

    frf_comparison_software.png

    Figure 21: Comparison of the computed FRF from the Fourier transform and using the modal software

    @@ -1041,8 +1041,8 @@ In the following analysis, FRF computed from the software will be used.
    -
    -

    6 Frequency Response Functions and Coherence Results

    +
    +

    6 Frequency Response Functions and Coherence Results

    Let's see one computed Frequency Response Function and one coherence in order to attest the quality of the measurement. @@ -1057,22 +1057,22 @@ First, we load the data.

    -And we plot on figure 22 the frequency response function from the force applied in the \(X\) direction at the location of the accelerometer number 11 to the acceleration in the \(X\) direction as measured by the first accelerometer located on the top platform of the hexapod. +And we plot on figure 22 the frequency response function from the force applied in the \(X\) direction at the location of the accelerometer number 11 to the acceleration in the \(X\) direction as measured by the first accelerometer located on the top platform of the hexapod.

    -The coherence associated is shown on figure 22. +The coherence associated is shown on figure 22.

    -
    +

    frf_result_example.png

    Figure 22: Example of one measured FRF

    -
    +

    coh_result_example.png

    Figure 23: Example of one measured Coherence

    @@ -1080,12 +1080,9 @@ The coherence associated is shown on figure 22.
    -
    -

    7 TODO Plot all coherences in one plot

    -
    -
    -

    8 Generation of a FRF matrix and a Coherence matrix from the measurements

    -
    +
    +

    7 Generation of a FRF matrix and a Coherence matrix from the measurements

    +

    We want here to combine all the Frequency Response Functions measured into one big array called the Frequency Response Matrix.

    @@ -1179,8 +1176,25 @@ And we save the obtained FRF matrix and Coherence matrix in a .mat
    -
    -

    9 Solid Bodies considered for further analysis

    +
    +

    8 Plot showing the coherence of all the measurements

    +
    +

    +Now that we have defined a Coherence matrix, we can plot each of its elements to have an idea of the overall coherence and thus, quality of the measurement. +The result is shown on figure 24. +

    + + +
    +

    all_coherence.png +

    +

    Figure 24: Plot of the coherence of all the measurements

    +
    +
    +
    + +
    +

    9 Solid Bodies considered for further analysis

    We consider the following solid bodies for further analysis: @@ -1195,7 +1209,7 @@ We consider the following solid bodies for further analysis:

    -We create a matlab structure solids that contains the accelerometers ID connected to each solid bodies (as shown on figure 8). +We create a matlab structure solids that contains the accelerometers ID connected to each solid bodies (as shown on figure 8). 

    solids = {};
@@ -1219,26 +1233,27 @@ Finally, we save that into a .mat file.
    -
    -

    10 Notes the solid body assumption

    + +
    +

    10 Note about the solid body assumption

    If we measure the motion of a rigid body along a direction \(\vec{x}\) using 2 sensors that are co-linear with the same direction \(\vec{x}\) (\(\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}\)), they will measured the same quantity.

    -This is illustrated on figure 24. +This is illustrated on figure 25.

    -
    +

    aligned_accelerometers.png

    -

    Figure 24: Aligned measurement of the motion of a solid body

    +

    Figure 25: Aligned measurement of the motion of a solid body

    -The motion of the rigid body of figure 24 is defined by the velocity \(\vec{v}\) and rotation \(\vec{\Omega}\) with respect to the reference frame \(\{O\}\). +The motion of the rigid body of figure 25 is defined by the velocity \(\vec{v}\) and rotation \(\vec{\Omega}\) with respect to the reference frame \(\{O\}\).

    @@ -1266,7 +1281,7 @@ However, we have \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\) because of the co-l

    -Two sensors that are measuring the co-linear motion of a rigid body should measure the same quantity. +Two sensors that are measuring the motion of a rigid body in the direction of the line linking the two sensors should measure the same quantity.

    @@ -1276,31 +1291,31 @@ We can verify that the rigid body assumption is correct by comparing the measure

    -From the table 2, we can guess which sensors will give the same results in the X and Y directions. +From the table 2, we can guess which sensors will give the same results in the X and Y directions.

    -Comparison of such measurements in the X direction is shown on figure 25 and in the Y direction on figure 26. +Comparison of such measurements in the X direction is shown on figure 26 and in the Y direction on figure 27.

    -
    +

    compare_acc_x_dir.png

    -

    Figure 25: Compare accelerometers align in the X direction

    +

    Figure 26: Compare accelerometers align in the X direction

    -
    +

    compare_acc_y_dir.png

    -

    Figure 26: Compare accelerometers align in the Y direction

    +

    Figure 27: Compare accelerometers align in the Y direction

    - +From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest.

    @@ -1309,7 +1324,7 @@ Comparison of such measurements in the X direction is shown on figure

    Author: Dehaeze Thomas

    -

    Created: 2019-07-04 jeu. 17:49

    +

    Created: 2019-07-05 ven. 11:46

    Validate

    diff --git a/modal-analysis/measurement.org b/modal-analysis/measurement.org index c684efe..9b3c648 100644 --- a/modal-analysis/measurement.org +++ b/modal-analysis/measurement.org @@ -812,7 +812,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp figure; for i = 1:size(acc_i, 1) - subaxis(3, 3, i); + subplot(3, 3, i); hold on; plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) @@ -858,7 +858,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp figure; for i = 1:size(acc_i, 1) - subaxis(3, 3, i); + subplot(3, 3, i); hold on; plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))
    Table 3: Impact test parameters