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Amplifier Piezoelectric Actuator APA300ML - Test Bench

Table of Contents


This report is also available as a pdf.


The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.

This include:

apa300ML.png

Figure 1: Picture of the APA300ML

1 Model of an Amplified Piezoelectric Actuator and Sensor

Consider a schematic of the Amplified Piezoelectric Actuator in Figure 2.

apa_model_schematic.png

Figure 2: Amplified Piezoelectric Actuator Schematic

A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):

\begin{equation} F_a = g_a \cdot V_a \end{equation}

A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\):

\begin{equation} V_s = g_s \cdot dl \end{equation}

We wish here to experimental measure \(g_a\) and \(g_s\).

The block-diagram model of the piezoelectric actuator is then as shown in Figure 3.

apa-model-simscape-schematic.png

Figure 3: Model of the APA with Simscape/Simulink

2 Geometrical Measurements

The received APA are shown in Figure 4.

IMG_20210224_143500.jpg

Figure 4: Received APA

2.1 Measurement Setup

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

IMG_20210224_143809.jpg

Figure 5: Measurement Setup

2.2 Measurement Results

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.

Table 1: Estimated flatness
  Flatness \([\mu m]\)
APA 1 8.9
APA 2 3.1
APA 3 9.1
APA 4 3.0
APA 5 1.9
APA 6 7.1
APA 7 18.7

3 Electrical Measurements

The capacitance of the stacks is measure with the LCR-800 Meter (doc)

IMG_20210312_120337.jpg

Figure 6: LCR Meter used for the measurements

The excitation frequency is set to be 1kHz.

Table 2: Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz
  Sensor Stack Actuator Stacks
APA 1 5.10 10.03
APA 2 4.99 9.85
APA 3 1.72 5.18
APA 4 4.94 9.82
APA 5 4.90 9.66
APA 6 4.99 9.91
APA 7 4.85 9.85

There is clearly a problem with APA300ML number 3

4 Stiffness measurement

4.1 APA test

load('meas_stiff_apa_1_x.mat', 't', 'F', 'd');
figure;
plot(t, F)
%% Automatic Zero of the force
F = F - mean(F(t > 0.1 & t < 0.3));

%% Start measurement at t = 0.2 s
d = d(t > 0.2);
F = F(t > 0.2);
t = t(t > 0.2); t = t - t(1);
i_l_start = find(F > 0.3, 1, 'first');
[~, i_l_stop] = max(F);
F_l = F(i_l_start:i_l_stop);
d_l = d(i_l_start:i_l_stop);
fit_l = polyfit(F_l, d_l, 1);

% %% Reset displacement based on fit
% d = d - fit_l(2);
% fit_s(2) = fit_s(2) - fit_l(2);
% fit_l(2) = 0;

% %% Estimated Stroke
% F_max = fit_s(2)/(fit_l(1) - fit_s(1));
% d_max = fit_l(1)*F_max;
h^2/fit_l(1)
figure;
hold on;
plot(F,d,'k')
plot(F_l, d_l)
plot(F_l, F_l*fit_l(1) + fit_l(2), '--')

5 Stroke measurement

We here wish to estimate the stroke of the APA.

To do so, one side of the APA is fixed, and a displacement probe is located on the other side as shown in Figure 7.

Then, a voltage is applied on either one or two stacks using a DAC and a voltage amplifier.

Here are the documentation of the equipment used for this test bench:

CE0EF55E-07B7-461B-8CDB-98590F68D15B.jpeg

Figure 7: Bench to measured the APA stroke

5.1 Voltage applied on one stack

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.

apa_stroke_voltage_time.png

Figure 8: Applied voltage as a function of time

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

apa_stroke_time_1s.png

Figure 9: Displacement as a function of time for all the APA300ML

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.

apa_d_vs_V_1s.png

Figure 10: Displacement as a function of the applied voltage

We can clearly see from Figure 10 that there is a problem with the APA number 3.

5.2 Voltage applied on two stacks

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.

apa_stroke_time_2s.png

Figure 11: Displacement as a function of time for all the APA300ML

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.

apa_d_vs_V_2s.png

Figure 12: Displacement as a function of the applied voltage

5.3 Voltage applied on all three stacks

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

apa_d_vs_V_3s.png

Figure 13: Displacement as a function of the applied voltage

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

Table 3: Measured maximum stroke
  Stroke \([\mu m]\)
APA 1 373.2
APA 2 365.5
APA 3 181.7
APA 4 359.7
APA 5 361.5
APA 6 363.9
APA 7 358.4

6 Test-Bench Description

Here are the documentation of the equipment used for this test bench:

test_bench_apa_alone.png

Figure 14: Schematic of the Test Bench

7 Measurement Procedure

7.1 Stroke Measurement

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.

7.2 Stiffness Measurement

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:

\begin{equation} k = \frac{\delta m g}{\delta d} \end{equation}

7.3 Hysteresis measurement

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.

expected_hysteresis.png

Figure 15: Expected Hysteresis (poel10_explor_activ_hard_mount_vibrat)

7.4 Piezoelectric Actuator Constant

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:

\begin{equation} d = g_{d/V_a} \cdot V_a \end{equation}

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 \(d\):

\begin{equation} d = g_{d/F_a} \cdot F_a \end{equation}

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

\begin{equation} g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} \end{equation}

7.5 Piezoelectric Sensor Constant

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

\begin{equation} V_s = g_{V_s/V_a} V_a \end{equation}

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 \(dl\):

\begin{equation} dl = g_{dl/F_a} F_a \end{equation}

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

\begin{equation} g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} \end{equation}

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.

7.6 Capacitance Measurement

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

7.7 Dynamical Behavior

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.

7.8 Compare the results obtained for all 7 APA300ML

Compare all the obtained parameters for all the test APA.

8 Measurement Results

9 Test Bench APA300ML - Simscape Model

9.1 Introduction

9.2 Nano Hexapod object

n_hexapod = struct();

9.2.1 APA - 2 DoF

n_hexapod.actuator = struct();

n_hexapod.actuator.type = 1;

n_hexapod.actuator.k  = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]

n_hexapod.actuator.c  = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]

n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

9.2.2 APA - Flexible Frame

n_hexapod.actuator.type = 2;

n_hexapod.actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]

n_hexapod.actuator.ks = 235e6; % Stiffness of one stack [N/m]
n_hexapod.actuator.cs = 1e1; % Stiffness of one stack [N/m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

9.2.3 APA - Fully Flexible

n_hexapod.actuator.type = 3;

n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

9.3 Identification

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'test_bench_apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'],  1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'],   1, 'openoutput'); io_i = io_i + 1; % Vertical Motion

%% Run the linearization
Ga = linearize(mdl, io, 0.0, options);
Ga.InputName  = {'Va'};
Ga.OutputName = {'Vs', 'dL', 'z'};

9.4 Compare 2-DoF with flexible

9.4.1 APA - 2 DoF

n_hexapod = struct();

n_hexapod.actuator = struct();

n_hexapod.actuator.type = 1;

n_hexapod.actuator.k  = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]

n_hexapod.actuator.c  = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]

n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]

n_hexapod.actuator.Ga = ones(6,1)*-2.15; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*2.305e-08; % Sensor gain [V/m]
G_2dof = linearize(mdl, io, 0.0, options);
G_2dof.InputName  = {'Va'};
G_2dof.OutputName = {'Vs', 'dL', 'z'};

9.4.2 APA - Fully Flexible

n_hexapod = struct();

n_hexapod.actuator.type = 3;

n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
G_flex = linearize(mdl, io, 0.0, options);
G_flex.InputName  = {'Va'};
G_flex.OutputName = {'Vs', 'dL', 'z'};

9.4.3 Comparison

10 Test Bench Struts - Simscape Model

10.1 Introduction

10.2 Nano Hexapod object

n_hexapod = struct();

10.2.1 Flexible Joint - Bot

n_hexapod.flex_bot = struct();

n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof

n_hexapod.flex_bot.kRx = ones(6,1)*5; % X bending stiffness [Nm/rad]
n_hexapod.flex_bot.kRy = ones(6,1)*5; % Y bending stiffness [Nm/rad]
n_hexapod.flex_bot.kRz = ones(6,1)*260; % Torsionnal stiffness [Nm/rad]
n_hexapod.flex_bot.kz  = ones(6,1)*1e8; % Axial stiffness [N/m]

n_hexapod.flex_bot.cRx = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cRy = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cRz = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cz  = ones(6,1)*1e2; %[N/(m/s)]

10.2.2 Flexible Joint - Top

n_hexapod.flex_top = struct();

n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof

n_hexapod.flex_top.kRx = ones(6,1)*5; % X bending stiffness [Nm/rad]
n_hexapod.flex_top.kRy = ones(6,1)*5; % Y bending stiffness [Nm/rad]
n_hexapod.flex_top.kRz = ones(6,1)*260; % Torsionnal stiffness [Nm/rad]
n_hexapod.flex_top.kz  = ones(6,1)*1e8; % Axial stiffness [N/m]

n_hexapod.flex_top.cRx = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cRy = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cRz = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cz  = ones(6,1)*1e2; %[N/(m/s)]

10.2.3 APA - 2 DoF

n_hexapod.actuator = struct();

n_hexapod.actuator.type = 1;

n_hexapod.actuator.k  = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]

n_hexapod.actuator.c  = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]

n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

10.2.4 APA - Flexible Frame

n_hexapod.actuator.type = 2;

n_hexapod.actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]

n_hexapod.actuator.ks = 235e6; % Stiffness of one stack [N/m]
n_hexapod.actuator.cs = 1e1; % Stiffness of one stack [N/m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

10.2.5 APA - Fully Flexible

n_hexapod.actuator.type = 3;

n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]

n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]

10.3 Identification

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'test_bench_struts';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'],  1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'],   1, 'openoutput'); io_i = io_i + 1; % Vertical Motion

%% Run the linearization
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName  = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};

10.4 Compare flexible joints

10.4.1 Perfect

n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
Gp = linearize(mdl, io, 0.0, options);
Gp.InputName  = {'Va'};
Gp.OutputName = {'Vs', 'dL', 'z'};

10.4.2 Top Flexible

n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
Gt = linearize(mdl, io, 0.0, options);
Gt.InputName  = {'Va'};
Gt.OutputName = {'Vs', 'dL', 'z'};

10.4.3 Bottom Flexible

n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
Gb = linearize(mdl, io, 0.0, options);
Gb.InputName  = {'Va'};
Gb.OutputName = {'Vs', 'dL', 'z'};

10.4.4 Both Flexible

n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
Gf = linearize(mdl, io, 0.0, options);
Gf.InputName  = {'Va'};
Gf.OutputName = {'Vs', 'dL', 'z'};

10.4.5 Comparison

11 Resonance frequencies - APA300ML

11.1 Introduction

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)

mode_bending_x.gif

Figure 16: X-bending mode (189Hz)

mode_bending_y.gif

Figure 17: Y-bending mode (285Hz)

mode_torsion_z.gif

Figure 18: Z-torsion mode (400Hz)

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.

11.2 Setup

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.

  • Laser Doppler Vibrometer Polytec OFV512
  • Instrumented hammer

measurement_setup_torsion.png

Figure 19: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer

11.3 Bending - X

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.

measurement_setup_X_bending.png

Figure 20: X-Bending measurement setup

The data is loaded.

bending_X = load('apa300ml_bending_X_top.mat');

The config for tfestimate is performed:

Ts = bending_X.Track1_X_Resolution; % Sampling frequency [Hz]
win = hann(ceil(1/Ts));

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

[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).

apa300ml_meas_freq_bending_x.png

Figure 21: Obtained FRF for the X-bending

11.4 Bending - Y

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).

measurement_setup_Y_bending.png

Figure 22: Y-Bending measurement setup

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.

apa300ml_meas_freq_bending_y.png

Figure 23: Obtained FRF for the Y-bending

11.5 Torsion - Z

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.

measurement_setup_torsion_bis.png

Figure 24: Z-Torsion measurement setup

The data is loaded, and the transfer function computed.

torsion = load('apa300ml_torsion_left.mat');
[G_torsion, ~]  = tfestimate(torsion.Track1, torsion.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?).

apa300ml_meas_freq_torsion_z.png

Figure 25: Obtained FRF for the Z-torsion

In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.

torsion = load('apa300ml_torsion_top.mat');
[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);

The two FRF are compared in Figure 26. It is clear that the first two modes does not correspond to the torsional mode. Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.

apa300ml_meas_freq_torsion_z_comp.png

Figure 26: Obtained FRF for the Z-torsion

11.6 Compare

The three measurements are shown in Figure 27.

apa300ml_meas_freq_compare.png

Figure 27: Obtained FRF - Comparison

11.7 Conclusion

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?).

Table 4: Measured frequency of the modes
Mode Strut Mode Measured Frequency
X-Bending 189Hz 280Hz
Y-Bending 285Hz 410Hz
Z-Torsion 400Hz ?

Bibliography

Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” CEAS Space Journal 10 (2). Springer:157–65.

Author: Dehaeze Thomas

Created: 2021-05-06 jeu. 21:55