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Rename footnotes

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Thomas Dehaeze 2025-02-04 14:23:26 +01:00
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@ -221,8 +221,8 @@ Finally, in Section ref:ssec:test_apa_spurious_resonances, the flexible modes of
<<ssec:test_apa_geometrical_measurements>>
To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of four points on each of the APA300ML interfaces.
From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:4] a plane through all the points.
As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:test_apa_3] is used to measure the height of four points on each of the APA300ML interfaces.
From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:test_apa_4] a plane through all the points.
The measured flatness values, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications.
#+begin_src matlab
@ -297,7 +297,7 @@ data2orgtable(1e6*apa_d', {'APA 1', 'APA 2', 'APA 3', 'APA 4', 'APA 5', 'APA 6',
From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$.
The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:1] shown in Figure ref:fig:test_apa_lcr_meter.
The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:test_apa_1] shown in Figure ref:fig:test_apa_lcr_meter.
The two stacks used as the actuator and the stack used as the force sensor were measured separately.
The measured capacitance values are summarized in Table ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$.
However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance.
@ -334,9 +334,9 @@ This may be because the manufacturer measures the capacitance with large signals
** Stroke and Hysteresis Measurement
<<ssec:test_apa_stroke_measurements>>
To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:test_apa_2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:12] and a voltage amplifier[fn:13].
The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:test_apa_12] and a voltage amplifier[fn:test_apa_13].
Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage).
#+name: fig:test_apa_stroke_bench
@ -412,7 +412,7 @@ exportFig('figs/test_apa_stroke_hysteresis.pdf', 'width', 'half', 'height', 'nor
In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
To experimentally estimate these modes, the APA is fixed at one end (see Figure ref:fig:test_apa_meas_setup_modes).
A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:7] is used to excite the flexible modes.
A Laser Doppler Vibrometer[fn:test_apa_6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:test_apa_7] is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software, and the results are shown in Figure ref:fig:test_apa_mode_shapes.
@ -520,7 +520,7 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
After the measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML.
This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing.
Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
An encoder[fn:8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
An encoder[fn:test_apa_8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
#+name: fig:test_bench_apa
#+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML
@ -577,7 +577,7 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a
<<ssec:test_apa_hysteresis>>
Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
A quasi static[fn:9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC.
A quasi static[fn:test_apa_9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC.
For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis.
This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]].
@ -1198,7 +1198,7 @@ for i = 1:length(i_kept)
end
#+end_src
The identified dynamics were then fitted by second order transfer functions[fn:10].
The identified dynamics were then fitted by second order transfer functions[fn:test_apa_10].
A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure ref:fig:test_apa_identified_damped_plants for different gains $g$.
It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
@ -1463,7 +1463,7 @@ Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$.
Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force.
Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,Ns/m$.
Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:5].
Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:test_apa_5].
Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series.
In this case, the total stiffness of the APA model is described by eqref:eq:test_apa_2dof_stiffness.
@ -1631,7 +1631,7 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
<<sec:test_apa_model_flexible>>
**** Introduction :ignore:
In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11].
In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:test_apa_11].
It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof.
This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape.
Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in Simscape as shown at the right by the "frames" =F1= to =F5=.
@ -2133,16 +2133,16 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
* Footnotes
[fn:13]PD200 from PiezoDrive. The gain is $20\,V/V$
[fn:12]The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution
[fn:11]Ansys\textsuperscript{\textregistered} was used
[fn:10]The transfer function fitting was computed using the =vectfit3= routine, see [[cite:&gustavsen99_ration_approx_frequen_domain_respon]]
[fn:9]Frequency of the sinusoidal wave is $1\,\text{Hz}$
[fn:8]Renishaw Vionic, resolution of $2.5\,nm$
[fn:7]Kistler 9722A
[fn:6]Polytec controller 3001 with sensor heads OFV512
[fn:5]Note that this is not completely correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.
[fn:4]The Matlab =fminsearch= command is used to fit the plane
[fn:3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$
[fn:2]Millimar 1318 probe, specified linearity better than $1\,\mu m$
[fn:1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$
[fn:test_apa_13]PD200 from PiezoDrive. The gain is $20\,V/V$
[fn:test_apa_12]The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution
[fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used
[fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see [[cite:&gustavsen99_ration_approx_frequen_domain_respon]]
[fn:test_apa_9]Frequency of the sinusoidal wave is $1\,\text{Hz}$
[fn:test_apa_8]Renishaw Vionic, resolution of $2.5\,nm$
[fn:test_apa_7]Kistler 9722A
[fn:test_apa_6]Polytec controller 3001 with sensor heads OFV512
[fn:test_apa_5]Note that this is not completely correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.
[fn:test_apa_4]The Matlab =fminsearch= command is used to fit the plane
[fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$
[fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\mu m$
[fn:test_apa_1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$