Measurement Analysis

1. Importation of the data
2. Variables for analysis
3. Measurement 1 - Effect of Ty stage
4. Measurement 2 - Effect of Ry stage
5. Measurement 3 - Effect of the Hexapod
6. Measurement 4 - Effect of the Splip-Ring and Spindle
7. Measurement 5 - Transmission from ground to marble

1 Importation of the data

First, load all the measurement files:

meas = {};
meas{1} = load('./mat/Measurement1.mat');
meas{2} = load('./mat/Measurement2.mat');
meas{3} = load('./mat/Measurement3.mat');
meas{4} = load('./mat/Measurement4.mat');
meas{5} = load('./mat/Measurement5.mat');

Change the track name for measurements 3 and 4.

meas{3}.Track1_Name = 'Input 1: Hexa Z';
meas{4}.Track1_Name = 'Input 1: Hexa Z';

For the measurements 1 to 4, the measurement channels are shown table 1.

Table 1: Channels for measurements 1 to 4
	Channel 1	Channel 2	Channel 3
Meas. 1	Input 1: tilt1 Z	Input 2: tilt2 Z	Input 3: Ty Y
Meas. 2	Input 1: tilt1 Z	Input 2: tilt2 Z	Input 3: Ty Y
Meas. 3	Input 1: Hexa Z	Input 2: tilt2 Z	Input 3: Ty Y
Meas. 4	Input 1: Hexa Z	Input 2: tilt2 Z	Input 3: Ty Y

For the measurement 5, the channels are shown table 2.

Table 2: Channels for measurement 5
	Channel 1	Channel 2	Channel 3	Channel 4
Meas. 5	Input 1: Floor Z	Input 2: Marble Z	Input 3: Floor Y	Input 4: Marble Y

When using two geophone sensors on the same tilt stage (measurements 1 and 2), we post-process the data to obtain the z displacement and the rotation of the tilt stage:

meas1_z    = (meas{1}.Track1+meas{1}.Track2)/2;
meas1_tilt = (meas{1}.Track1-meas{1}.Track2)/2;
meas{1}.Track1 = meas1_z;
meas{1}.Track1_Y_Magnitude = 'Meter / second';
meas{1}.Track1_Name = 'Ry Z';
meas{1}.Track2 = meas1_tilt;
meas{1}.Track2_Y_Magnitude = 'Rad / second';
meas{1}.Track2_Name = 'Ry Tilt';

meas2_z    = (meas{2}.Track1+meas{2}.Track2)/2;
meas2_tilt = (meas{2}.Track1-meas{2}.Track2)/2;
meas{2}.Track1 = meas2_z;
meas{2}.Track1_Y_Magnitude = 'Meter / second';
meas{2}.Track1_Name = 'Ry Z';
meas{2}.Track2 = meas2_tilt;
meas{2}.Track2_Y_Magnitude = 'Rad / second';
meas{2}.Track2_Name = 'Ry Tilt';

2 Variables for analysis

We define the sampling frequency and the time vectors for the plots.

Fs = 256; % [Hz]
dt = 1/(Fs);
t1  = dt*[1:length(meas{1}.Track1)];
t2  = dt*[1:length(meas{2}.Track1)];
t3  = dt*[1:length(meas{3}.Track1)];
t4  = dt*[1:length(meas{4}.Track1)];
t5  = dt*[1:length(meas{5}.Track1)];

For the frequency analysis, we define the frequency limits for the plot.

fmin = 1; % [Hz]
fmax = 100; % [Hz]

Then we define the windows that will be used to average the results.

psd_window = hanning(2*fmin/dt);

3 Measurement 1 - Effect of Ty stage

The configuration for this measurement is shown table 3.

Table 3: Stages configuration - Measurement 1
Time	0-309	309-end
Ty	OFF	ON

We then plot the measurements in time domain (figure 1).

We observe strange behavior when the Ty stage is turned on. How can we explain that?

Figure 1: Time domain - measurement 1

To understand what is going on, instead of looking at the velocity, we can look at the displacement by integrating the data. The displacement is computed by integrating the velocity using cumtrapz function.

tdisp = t1(ceil(300/dt):ceil(340/dt));
xdisp = cumtrapz(tdisp, meas{1}.Track3(ceil(300/dt):ceil(340/dt)));

Then we plot the position with respect to time (figure 2).

Figure 2: Y displacement of the Ty stage

We when compute the power spectral density of each measurement before and after turning on the stage.

[pxx111, f111] = pwelch(meas{1}.Track1(1:ceil(300/dt)),   psd_window, [], [], Fs);
[pxx112, f112] = pwelch(meas{1}.Track1(ceil(350/dt):end), psd_window, [], [], Fs);

[pxx121, f121] = pwelch(meas{1}.Track2(1:ceil(300/dt)),   psd_window, [], [], Fs);
[pxx122, f122] = pwelch(meas{1}.Track2(ceil(350/dt):end), psd_window, [], [], Fs);

[pxx131, f131] = pwelch(meas{1}.Track3(1:ceil(300/dt)),   psd_window, [], [], Fs);
[pxx132, f132] = pwelch(meas{1}.Track3(ceil(350/dt):end), psd_window, [], [], Fs);

We finally plot the power spectral density of each track (figures 3, 4, 5).

Figure 3: PSD of the Z velocity of Ry stage - measurement 1

Figure 4: PSD of the Rotation of Ry Stage - measurement 1

Figure 5: PSD of the Ty velocity in the Y direction - measurement 1

Turning on the Y-translation stage increases the velocity of the Ty stage in the Y direction and the rotation motion of the tilt stage:

at 20Hz
at 40Hz
between 80Hz and 90Hz

It does not seems to have any effect on the Z motion of the tilt stage.

4 Measurement 2 - Effect of Ry stage

The tilt stage is turned ON at around 326 seconds (table 4).

Table 4: Stages configuration - Measurement 2
Time	0-326	326-end
Tilt	OFF	ON

We plot the time domain (figure 6) and we don't observe anything special in the time domain.

Figure 6: Time domain - measurement 2

We compute the PSD of each track and we plot them (figures 7, 8 and 9 ).

[pxx211, f211] = pwelch(meas{2}.Track1(1:ceil(326/dt)),   psd_window, [], [], Fs);
[pxx212, f212] = pwelch(meas{2}.Track1(ceil(326/dt):end), psd_window, [], [], Fs);

[pxx221, f221] = pwelch(meas{2}.Track2(1:ceil(326/dt)),   psd_window, [], [], Fs);
[pxx222, f222] = pwelch(meas{2}.Track2(ceil(326/dt):end), psd_window, [], [], Fs);

[pxx231, f231] = pwelch(meas{2}.Track3(1:ceil(326/dt)),   psd_window, [], [], Fs);
[pxx232, f232] = pwelch(meas{2}.Track3(ceil(326/dt):end), psd_window, [], [], Fs);

Figure 7: PSD of the Z velocity of Ry Stage - measurement 2

Figure 8: PSD of the Rotation motion of Ry Stage - measurement 2

Figure 9: PSD of the Ty velocity in the Y direction - measurement 2

We observe no noticeable difference when the Tilt-stage is turned ON expect a small decrease of the Z motion of the tilt stage around 10Hz.

5 Measurement 3 - Effect of the Hexapod

The hexapod is turned off after 406 seconds (table 5).

Table 5: Stages configuration - Measurement 3
Time	0-406	406-end
Tilt	ON	ON
Hexa	ON	OFF

The time domain result is shown figure 10.

Figure 10: Time domain - measurement 3

We then compute the PSD of each track before and after turning off the hexapod and plot the results in the figures 11, 12 and 13.

[pxx311, f311] = pwelch(meas{3}.Track1(1:ceil(400/dt)),   psd_window, [], [], Fs);
[pxx312, f312] = pwelch(meas{3}.Track1(ceil(420/dt):end), psd_window, [], [], Fs);

[pxx321, f321] = pwelch(meas{3}.Track2(1:ceil(400/dt)),   psd_window, [], [], Fs);
[pxx322, f322] = pwelch(meas{3}.Track2(ceil(420/dt):end), psd_window, [], [], Fs);

[pxx331, f331] = pwelch(meas{3}.Track3(1:ceil(400/dt)),   psd_window, [], [], Fs);
[pxx332, f332] = pwelch(meas{3}.Track3(ceil(420/dt):end), psd_window, [], [], Fs);

Figure 11: PSD of the Z velocity of the Hexapod - measurement 3

Figure 12: PSD of the Z velocity of the Ry stage - measurement 3

Figure 13: PSD of the Ty velocity in the Y direction - measurement 3

Turning ON induces some motion on the hexapod in the z direction (figure 11), on the tilt stage in the z direction (figure 12) and on the y motion of the Ty stage (figure 13):

at 17Hz
at 34Hz

6 Measurement 4 - Effect of the Splip-Ring and Spindle

The slip ring is turned on at 300s, then the spindle is turned on at 620s (table 6). The time domain signals are shown figure 14.

Table 6: Stages configuration - Measurement 4
Time	0-300	300-620	620-end
SlipRing	OFF	ON	ON
Hexa	OFF	OFF	OFF
Spindle	OFF	OFF	ON

Figure 14: Time domain - measurement 4

The PSD of each track are computed using the code below.

Figure 15: PSD of the Z velocity of the Hexapod - measurement 4

Figure 16: PSD of the Ry rotation in the Y direction - measurement 4

Figure 17: PSD of the Ty velocity in the Y direction - measurement 4

Turning ON the splipring seems to not add motions on the stages measured. It even seems to lower the motion of the Ty stage (figure 17): does that make any sense?

Turning ON the spindle induces motions:

at 5Hz on each motion measured
at 22.5Hz on the Z motion of the Hexapod. Can this is due to some 50Hz?
at 62Hz on each motion measured

7 Measurement 5 - Transmission from ground to marble

This measurement just consists of measurement of Y-Z motion of the ground and the marble.

The time domain signals are shown on figure 18.

Figure 18: Time domain - measurement 5

We compute the PSD of each track and we plot the PSD of the Z motion for the ground and marble on figure 19 and for the Y motion on figure 20.

[pxx51, f51] = pwelch(meas{5}.Track1(:), psd_window, [], [], Fs);
[pxx52, f52] = pwelch(meas{5}.Track2(:), psd_window, [], [], Fs);
[pxx53, f53] = pwelch(meas{5}.Track3(:), psd_window, [], [], Fs);
[pxx54, f54] = pwelch(meas{5}.Track4(:), psd_window, [], [], Fs);

Figure 19: PSD of the ground and marble in the Z direction

Figure 20: PSD of the ground and marble in the Y direction

Then, instead of looking at the Power Spectral Density, we can try to estimate the transfer function from a ground motion to the motion of the marble. The transfer functions are shown on figure 21 and the coherence on figure 22.

[tfz, fz] = tfestimate(meas{5}.Track1(:), meas{5}.Track2(:), psd_window, [], [], Fs);
[tfy, fy] = tfestimate(meas{5}.Track3(:), meas{5}.Track4(:), psd_window, [], [], Fs);

Figure 21: Transfer function estimation - measurement 5

[cohz, fz] = mscohere(meas{5}.Track1(:), meas{5}.Track2(:), psd_window, [], [], Fs);
[cohy, fy] = mscohere(meas{5}.Track3(:), meas{5}.Track4(:), psd_window, [], [], Fs);

Figure 22: Coherence - measurement 5

The marble seems to have a resonance at around 20Hz on the Y direction. But the coherence is not good above 20Hz, so it is difficult to estimate resonances.