UP | HOME

Modal Analysis - Measurement

Table of Contents

All the files (data and Matlab scripts) are accessible here.

1 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)

oros.png

Figure 1: Acquisition system: OROS

The acquisition system permits to auto-range the inputs (probably using variable gain amplifiers) the obtain the maximum dynamic range. This is done before each measurement. Anti-aliasing filters are also included in the system.

accelero_M393B05.png

Figure 2: Accelerometer used: M393B05

Table 1: 393B05 Accelerometer Data Sheet
Sensitivity 10V/g
Measurement Range 0.5 g pk
Broadband Resolution 0.000004 g rms
Frequency Range 0.7 to 450Hz
Resonance Frequency > 2.5kHz

Tests have been conducted to determine the most suitable Hammer tip. This has been found that the softer tip gives the best results. It excites more the low frequency range where the coherence is low, the overall coherence was improved.

instrumented_hammer.png

Figure 3: Instrumented Hammer

hammer_tips.png

Figure 4: Hammer tips

The accelerometers are glued on the structure.

2 Structure Preparation and Test Planning

2.1 Structure Preparation

All the stages are turned ON. This is done for two reasons:

  • Be closer to the real dynamic of the station in used
  • If the control system of stages are turned OFF, this would results in very low frequency modes un-identifiable with the current setup, and this will also decouple the dynamics which would not be the case in practice

This is critical for the translation stage and the spindle as their is no stiffness in the free DOF (air-bearing for the spindle for instance).

The alternative would have been to mechanically block the stages with screws, but this may result in changing the modes.

The stages turned ON are:

  • Translation Stage
  • Tilt Stage
  • Spindle and Slip-Ring
  • Hexapod

The top part representing the NASS and the sample platform have been removed in order to reduce the complexity of the dynamics and also because this will be further added in the model inside Simscape.

All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod).

All other elements have been remove from the granite such as another heavy positioning system.

2.2 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).

However, the principle of reciprocity states that: \[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \] Thus, only one column or one line of the matrix has to be identified.

Either we choose to identify \(\frac{X_k}{F_i}\) or \(\frac{X_i}{F_k}\) for any chosen \(k\) and for \(i = 1,\ ...,\ N\).

We here choose to identify \(\frac{X_i}{F_k}\) for practical reasons:

  • it is easier to glue the accelerometers on some stages than to excite this particular stage with the Hammer

The measurement thus consists of:

  • always excite the structure at the same location with the Hammer
  • Move the accelerometers to measure all the DOF of the structure

2.3 Location of the Accelerometers

4 tri-axis accelerometers are used for each solid body.

Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some redundancy.

This could also help us identify measurement problems or flexible modes is present.

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 top of the tilt stage
  • 3 on top of the spindle
  • 4 on top of the hexapod (figure 7)

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 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

The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) is shown below. The values are in meter. They are contained in the mat/id31_nanostation.cfg file.

ID X[m] Y[m] Z[m]
1 -6.4000e-002 -6.4000e-002 -2.7000e-001
2 -6.4000e-002 6.4000e-002 -2.7000e-001
3 6.4000e-002 6.4000e-002 -2.7000e-001
4 6.4000e-002 -6.4000e-002 -2.7000e-001
5 -3.8500e-001 -3.0000e-001 -4.1680e-001
6 -4.2000e-001 2.8000e-001 -4.1680e-001
7 4.2000e-001 2.8000e-001 -4.1680e-001
8 3.8000e-001 -3.0000e-001 -4.1680e-001
9 -4.7500e-001 -4.1400e-001 -4.2730e-001
10 -4.6500e-001 4.0700e-001 -4.2730e-001
11 4.7500e-001 4.2400e-001 -4.2730e-001
12 4.7500e-001 -4.1900e-001 -4.2730e-001
13 -3.2000e-001 -4.4600e-001 -7.8560e-001
14 -4.8000e-001 5.3400e-001 -7.8560e-001
15 4.5000e-001 5.3400e-001 -7.8560e-001
16 2.9500e-001 -4.8100e-001 -7.8560e-001
17 -7.3000e-001 -5.2600e-001 -9.5060e-001
18 -7.3500e-001 8.1400e-001 -9.5060e-001
19 8.7500e-001 7.9900e-001 -9.5060e-001
20 8.6490e-001 -5.0600e-001 -9.5060e-001
21 -1.5500e-001 -9.0000e-002 -5.9400e-001
22 0.0000e+000 1.8000e-001 -5.9400e-001
23 1.5500e-001 -9.0000e-002 -5.9400e-001

2.4 Hammer Impacts

Only 3 impact points are used.

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

We chose this excitation point as it seems to excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.

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

3 Signal Processing

3.1 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 2.

Table 2: Impact test parameters
Start Delay -5%
Number of lines 801 lines
Trigger Threshold 300N
Trigger Hysteresis 100mN
FRF Average 10

3.2 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. The parameters are:

  • Start: \(3\%\)
  • Stop: \(7\%\)

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. The parameters are:

  • FlatTop:
    • Start: \(3\%\)
    • Stop: \(2.96\%\)
  • Decreasing point:
    • X: \(60.4\%\)
    • Y: \(14.7\%\)

windowing_response_signal.png

Figure 14: Window used for the response signals

4 Force and Response signals

Let's load some obtained data to look at the force and response signals. 

meas1_raw = load('mat/meas_raw_1.mat');

4.1 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 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

4.2 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 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

4.3 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).

The slight difference can probably be explained by the use of windows.

In the following analysis, FRF computed from the software will be used. 

meas1 = load('mat/meas_frf_coh_1.mat');

frf_comparison_software.png

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

5 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.

First, we load the data. 

meas1 = load('mat/meas_frf_coh_1.mat');

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.

frf_result_example.png

Figure 22: Example of one measured FRF

coh_result_example.png

Figure 23: Example of one measured Coherence

6 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.

The frequency response matrix is an \(n \times p \times q\):

  • \(n\) is the number of measurements: \(23 \times 3\) (23 accelerometers measuring 3 directions each)
  • \(p\) is the number of excitation inputs: \(3\)
  • \(q\) is the number of frequency points \(\omega_i\)

Thus, the FRF matrix is an \(69 \times 3 \times 801\) matrix.

For each frequency point \(\omega_i\), we obtain a 2D matrix:

\begin{equation} \text{FRF}(\omega_i) = \begin{bmatrix} \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\ \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\ \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\ \frac{D_{2_x}}{F_x}(\omega_i) & \frac{D_{2_x}}{F_y}(\omega_i) & \frac{D_{2_x}}{F_z}(\omega_i) \\ \vdots & \vdots & \vdots \\ \frac{D_{23_z}}{F_x}(\omega_i) & \frac{D_{23_z}}{F_y}(\omega_i) & \frac{D_{23_z}}{F_z}(\omega_i) \\ \end{bmatrix} \end{equation}

We generate such FRF matrix from the measurements using the following script. 

n_meas = 24;
n_acc = 23;

dirs = 'XYZ';

% Number of Accelerometer * DOF for each acccelerometer / Number of excitation / frequency points
FRFs = zeros(3*n_acc, 3, 801);
COHs = zeros(3*n_acc, 3, 801);

% Loop through measurements
for i = 1:n_meas
  % Load the measurement file
  meas = load(sprintf('mat/meas_frf_coh_%i.mat', i));

  % First: determine what is the exitation (direction and sign)
  exc_dir = meas.FFT1_AvXSpc_2_1_RMS_RfName(end);
  exc_sign = meas.FFT1_AvXSpc_2_1_RMS_RfName(end-1);
  % Determine what is the correct excitation sign
  exc_factor = str2num([exc_sign, '1']);
  if exc_dir ~= 'Z'
    exc_factor = exc_factor*(-1);
  end

  % Then: loop through the nine measurements and store them at the correct location
  for j = 2:10
    % Determine what is the accelerometer and direction
    [indices_acc_i] = strfind(meas.(sprintf('FFT1_H1_%i_1_RpName', j)), '.');
    acc_i = str2num(meas.(sprintf('FFT1_H1_%i_1_RpName', j))(indices_acc_i(1)+1:indices_acc_i(2)-1));

    meas_dir  = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end);
    meas_sign = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end-1);
    % Determine what is the correct measurement sign
    meas_factor = str2num([meas_sign, '1']);
    if meas_dir ~= 'Z'
      meas_factor = meas_factor*(-1);
    end

    % FRFs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j));
    % COHs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j));

    FRFs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j));
    COHs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j));
  end
end
freqs = meas.FFT1_Coh_10_1_RMS_X_Val;

And we save the obtained FRF matrix and Coherence matrix in a .mat file. 

save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');

Author: Dehaeze Thomas

Created: 2019-07-03 mer. 17:24

Validate