Amplifier Piezoelectric Actuator APA300ML - Test Bench

The flatness corresponding to the two interface planes are measured as shown in Figure 5.

The height (Z) measurements at the 8 locations (4 points by plane) are defined below.

apa1  = 1e-6*[0, -0.5 , 3.5  , 3.5  , 42   , 45.5, 52.5 , 46];
apa2  = 1e-6*[0, -2.5 , -3   , 0    , -1.5 , 1   , -2   , -4];
apa3  = 1e-6*[0, -1.5 , 15   , 17.5 , 6.5  , 6.5 , 21   , 23];
apa4  = 1e-6*[0, 6.5  , 14.5 , 9    , 16   , 22  , 29.5 , 21];
apa5  = 1e-6*[0, -12.5, 16.5 , 28.5 , -43  , -52 , -22.5, -13.5];
apa6  = 1e-6*[0, -8   , -2   , 5    , -57.5, -62 , -55.5, -52.5];
apa7  = 1e-6*[0, 19.5 , -8   , -29.5, 75   , 97.5, 70   , 48];
apa7b = 1e-6*[0, 9    , -18.5, -30  , 31   , 46.5, 16.5 , 7.5];
apa = {apa1, apa2, apa3, apa4, apa5, apa6, apa7b};

The X/Y Positions of the 8 measurement points are defined below.

W = 20e-3;   % Width [m]
L = 61e-3;   % Length [m]
d = 1e-3;    % Distance from border [m]
l = 15.5e-3; % [m]

pos = [[-L/2 + d; W/2 - d], [-L/2 + l - d; W/2 - d], [-L/2 + l - d; -W/2 + d], [-L/2 + d; -W/2 + d], [L/2 - l + d; W/2 - d], [L/2 - d; W/2 - d], [L/2 - d; -W/2 + d], [L/2 - l + d; -W/2 + d]];

Finally, the flatness is estimated by fitting a plane through the 8 points using the fminsearch command.

apa_d = zeros(1, 7);
for i = 1:7
    fun = @(x)max(abs(([pos; apa{i}]-[0;0;x(1)])'*([x(2:3);1]/norm([x(2:3);1]))));
    x0 = [0;0;0];
    [x, min_d] = fminsearch(fun,x0);
    apa_d(i) = min_d;
end

The obtained flatness are shown in Table 1.

Let’s first look at the relation between the voltage applied to one stack to the displacement of the APA as measured by the displacement probe.

The applied voltage is shown in Figure 8.

The obtained displacement is shown in Figure 9. The displacement is set to zero at initial time when the voltage applied is -20V.

Finally, the displacement is shown as a function of the applied voltage in Figure 10. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Now look at the relation between the voltage applied to the two other stacks to the displacement of the APA as measured by the displacement probe.

The obtained displacement is shown in Figure 11. The displacement is set to zero at initial time when the voltage applied is -20V.

Finally, the displacement is shown as a function of the applied voltage in Figure 12. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Finally, we can combine the two measurements to estimate the relation between the displacement and the voltage applied to the three stacks (Figure 13).

The obtained maximum stroke for all the APA are summarized in Table 3.

Using the PD200 amplifier, output a voltage: \[ V_a = 65 + 85 \sin(2\pi \cdot t) \] To have a quasi-static excitation between -20 and 150V.

As the gain of the PD200 amplifier is 20, the DAC output voltage should be: \[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \]

Verify that the voltage offset of the PD200 is zero!

Measure the output vertical displacement \(d\) using the interferometer.

Then, plot \(d\) as a function of \(V_a\), and perform a linear regression. Conclude on the obtained stroke.

Add some (known) weight \(\delta m g\) on the suspended mass and measure the deflection \(\delta d\). This can be tested when the piezoelectric stacks are open-circuit.

As the stiffness will be around \(k \approx 10^6 N/m\), an added mass of \(m \approx 100g\) will induce a static deflection of \(\approx 1\mu m\) which should be large enough for a precise measurement using the interferometer.

Then the obtained stiffness is:

Supply a quasi static sinusoidal excitation \(V_a\) at different voltages.

The offset should be 65V, and the sin amplitude can range from 1V up to 85V.

For each excitation amplitude, the vertical displacement \(d\) of the mass is measured.

Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes.

Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\). Use a quasi static (1Hz) excitation signal \(V_a\) on the piezoelectric stack and measure the vertical displacement \(d\). Perform a linear regression to obtain:

Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\):

From the two gains, it is then easy to determine \(g_a\):

From a quasi static excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\):

Note here that there is an high pass filter formed by the piezo capacitor and parallel resistor. The excitation frequency should then be in between the cut-off frequency of this high pass filter and the first resonance.

Alternatively, the gain can be computed from the dynamical identification and taking the gain at the wanted frequency.

Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):

Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is:

Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. This external force can be some weight added, or a piezo in parallel.

Measure the capacitance of the 3 stacks individually using a precise multi-meter.

Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the generated voltage \(V_s\).

This can be performed using different excitation signals.

This can also be performed with and without the encoder fixed to the APA.

Compare all the obtained parameters for all the test APA.

Three main resonances are foreseen to be problematic for the control of the APA300ML:

• Mode in X-bending at 189Hz (Figure 16)
• Mode in Y-bending at 285Hz (Figure 17)
• Mode in Z-torsion at 400Hz (Figure 18)

These modes are present when flexible joints are fixed to the ends of the APA300ML.

In this section, we try to find the resonance frequency of these modes when one end of the APA is fixed and the other is free.

The measurement setup is shown in Figure 19. A Laser vibrometer is measuring the difference of motion of two points. The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.

The setup to measure the X-bending motion is shown in Figure 20. The APA is excited with an instrumented hammer having a solid metallic tip. The impact point is on the back-side of the APA aligned with the top measurement point.

The data is loaded.

bending_X = load('apa300ml_bending_X_top.mat')

Ts = bending_X.Track1_X_Resolution; % Sampling frequency [Hz]

The transfer function from the input force to the output “rotation” (difference between the two measured distances).

win = hann(ceil(1/Ts));
[G_bending_X, f]  = tfestimate(bending_X.Track1, bending_X.Track2, win, [], [], 1/Ts);

The result is shown in Figure 21.

The can clearly observe a nice peak at 280Hz, and then peaks at the odd “harmonics” (third “harmonic” at 840Hz, and fifth “harmonic” at 1400Hz).

The setup to measure the Y-bending is shown in Figure 22.

The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points).

The data is loaded, and the transfer function from the force to the measured rotation is computed.

bending_Y = load('apa300ml_bending_Y_top.mat')
[G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, [], [], 1/Ts);

The results are shown in Figure 23. The main resonance is at 412Hz, and we also see the third “harmonic” at 1220Hz.

Finally, we measure the Z-torsion resonance as shown in Figure 24.

The excitation is shown on the other side of the APA, on the side to excite the torsion motion.

The data is loaded, and the transfer function computed.

torsion = load('apa300ml_torsion_left.mat')
[G_torsion, ~]  = tfestimate(torsion_left.Track1, torsion_left.Track2, win, [], [], 1/Ts);

The results are shown in Figure 25. We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz. And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz. The mode in pure torsion is probably at higher frequency (peak around 1kHz?).

The three measurements are shown in Figure 26.

When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small).

In the current tests, the APA are in a fixed/free condition. Therefore, it is quite obvious that we measured higher resonance frequencies than what is foreseen for the struts. It is however quite interesting that there is a factor \(\approx \sqrt{2}\) between the two (increased of the stiffness by a factor 2?).