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18
nass-fem.bib
18
nass-fem.bib
@ -33,6 +33,15 @@
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@book{hatch00_vibrat_matlab_ansys,
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author = {Hatch, Michael R},
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title = {Vibration simulation using MATLAB and ANSYS},
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year = 2000,
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publisher = {CRC Press},
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}
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||||
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@phdthesis{rankers98_machin,
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author = {Rankers, Adrian Mathias},
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keywords = {favorite},
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@ -44,15 +53,6 @@
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@book{hatch00_vibrat_matlab_ansys,
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author = {Hatch, Michael R},
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title = {Vibration simulation using MATLAB and ANSYS},
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year = 2000,
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publisher = {CRC Press},
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}
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@article{craig68_coupl_subst_dynam_analy,
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author = {ROY R. CRAIG and MERVYN C. C. BAMPTON},
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title = {Coupling of Substructures for Dynamic Analyses.},
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344
nass-fem.org
344
nass-fem.org
@ -1149,8 +1149,6 @@ First, the fundamental principles and methodological approaches of this modeling
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It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section ref:ssec:detail_fem_super_element_example).
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Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section ref:ssec:detail_fem_super_element_validation).
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The work presented in this section has also been published in [[cite:&brumund21_multib_simul_reduc_order_flexib_bodies_fea]].
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@ -1213,7 +1211,7 @@ The presented modeling framework was first applied to an Amplified Piezoelectric
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Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section ref:sec:detail_fem_actuator.
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Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing.
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The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure ref:fig:detail_fem_apa95ml_picture) and an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement into the vertical direction [[cite:&claeyssen07_amplif_piezoel_actuat]].
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The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement into the vertical direction [[cite:&claeyssen07_amplif_piezoel_actuat]].
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The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework.
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The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation.
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@ -1233,8 +1231,8 @@ The specific design of the APA allows for the simultaneous modeling of a mechani
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| *Parameter* | *Value* |
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|----------------+---------------|
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| Nominal Stroke | $100\,\mu m$ |
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| Blocked force | $1600\,N$ |
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| Stiffness | $16\,N/\mu m$ |
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| Blocked force | $2100\,N$ |
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| Stiffness | $21\,N/\mu m$ |
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#+latex: \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
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#+end_minipage
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@ -1252,68 +1250,59 @@ The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was th
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| Stainless Steel | $190\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ |
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| Piezoelectric Ceramics (PZT) | $49.5\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ |
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The definition of interface frames, or "remote points" as depicted in Figure ref:fig:detail_fem_apa95ml_frames, constitute a critical aspect of the model preparation.
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Seven frames were established: two frames for each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation.
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The definition of interface frames, or "remote points", constitute a critical aspect of the model preparation.
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Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation.
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Six additional modes were considered, resulting in total model order of $48$.
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The modal reduction procedure was then executed, yielding the reduced mass and stiffness matrices that form the foundation of the component's representation in the multi-body simulation environment.
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#+name: fig:detail_fem_apa95ml_model
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#+caption: Finite element model of the APA95ML. Obtained mesh is shown in (\subref{fig:detail_fem_apa95ml_mesh}). Frames (or "remote points") used for the modal reduction are shown in (\subref{fig:detail_fem_apa95ml_frames}).
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#+caption: Obtained mesh and defined interface frames (or "remote points") in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_modal_schematic}).
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#+attr_latex: :options [htbp]
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#+begin_figure
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#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh}
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#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh and "remote points"}
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#+attr_latex: :options {0.48\textwidth}
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#+begin_subfigure
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#+attr_latex: :width 0.95\linewidth
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#+attr_latex: :scale 1
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[[file:figs/detail_fem_apa95ml_mesh.png]]
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#+end_subfigure
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#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_frames}Defined frames}
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#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model}
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#+attr_latex: :options {0.48\textwidth}
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#+begin_subfigure
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#+attr_latex: :width 0.85\linewidth
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[[file:figs/detail_fem_apa95ml_frames.png]]
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#+attr_latex: :scale 1
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[[file:figs/detail_fem_apa_modal_schematic.png]]
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#+end_subfigure
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#+end_figure
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**** Super Element in the Multi-Body Model
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Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called "Reduced Order Flexible Solid".
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This block has several interface frames corresponding to the ones defined in the FEA software.
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Frame $\{4\}$ was connected to the "world" frame, while frame $\{6\}$ was coupled to a vertically guided payload.
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In this example, two piezoelectric stacks were used for actuation while one piezoelectric stack was used as a force sensor.
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Therefore, a force source $F_a$ operating between frames $\{3\}$ and $\{2\}$ was used, while a displacement sensor $d_L$ between frames $\{1\}$ and $\{7\}$ was used for the sensor stack.
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This is illustrated in Figure ref:fig:detail_fem_apa_model_schematic.
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Model:
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- Connect frame $\{4\}$ to world frame and frame $\{6\}$ to a 5.5kg mass, vertically guided
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- 2 actuator stacks, 1 sensor stack:
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- force source between frames $\{3\}$ and $\{2\}$
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- measured strain for force sensor by measuring the displacement between $\{1\}$ and $\{7\}$
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- Input: internal force applied
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- Output: strain in the sensor stack
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- Issue: how to convert voltage to force and strain to voltage?
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#+name: fig:detail_fem_apa_model_schematic
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#+caption: Amplified Piezoelectric Actuator Schematic
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#+attr_latex: :width 0.5\linewidth
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[[file:figs/detail_fem_apa_modal_schematic.png]]
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Need to link the electrical domain (voltages, charges) with the mechanical domain (forces, strain).
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To do so, "actuator constant" $g_a$ and "sensor constant" $g_s$ are used as shown in Figure ref:fig:detail_fem_apa_model_schematic.
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A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$:
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\begin{equation}
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\boxed{F_a = g_a \cdot V_a}
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\end{equation}
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A change of length $dl$ of the sensor stack will induce a voltage $V_s$:
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\begin{equation}
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\boxed{V_s = g_s \cdot dl}
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\end{equation}
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|
||||
In order to correctly model the piezoelectric actuator with Simscape, the values for $g_a$ and $g_s$ needs to be determined.
|
||||
- $g_a$: the ratio of the generated force $F_a$ to the supply voltage $V_a$ across the piezoelectric stack
|
||||
- $g_s$: the ratio of the generated voltage $V_s$ across the piezoelectric stack when subject to a strain $\Delta h$
|
||||
However, to have access to the physical voltage input of the actuators stacks $V_a$ and to the generated voltage by the force sensor $V_s$, conversion between the electrical and mechanical domains need to be determined.
|
||||
|
||||
**** Sensor and Actuator "constants"
|
||||
|
||||
The gains $g_a$ and $g_s$ were estimated from the physical properties of the piezoelectric stack material (summarized in Table ref:tab:detail_fem_stack_parameters).
|
||||
To link the electrical domain to the mechanical domain, an "actuator constant" $g_a$ and a "sensor constant" $g_s$ were introduced as shown in Figure ref:fig:detail_fem_apa_model_schematic.
|
||||
|
||||
From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:detail_fem_dl_to_vs.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_dl_to_vs}
|
||||
V_s = g_s \cdot d_L, \quad g_s = \frac{d_{33}}{\epsilon^T s^D n}
|
||||
\end{equation}
|
||||
|
||||
From [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:detail_fem_va_to_fa.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_va_to_fa}
|
||||
F_a = g_a \cdot V_a, \quad g_a = d_{33} n k_a, \quad k_a = \frac{c^{E} A}{L}
|
||||
\end{equation}
|
||||
|
||||
Unfortunately, it is difficult to know exactly which material is used in the amplified piezoelectric actuator[fn:1].
|
||||
However, based on the available properties of the stacks in the data-sheet (summarized in Table ref:tab:detail_fem_stack_parameters), the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties.
|
||||
|
||||
#+name: tab:detail_fem_stack_parameters
|
||||
#+caption: Stack Parameters
|
||||
@ -1329,19 +1318,7 @@ The gains $g_a$ and $g_s$ were estimated from the physical properties of the pie
|
||||
| Length | $mm$ | 20 |
|
||||
| Stack Area | $mm^2$ | 10x10 |
|
||||
|
||||
From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force.
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
|
||||
F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Unfortunately, it is difficult to know exactly which material is used in the amplified piezoelectric actuator[fn:1].
|
||||
However, based on the available properties of the stacks in the data-sheet (summarized in Table ref:tab:detail_fem_stack_parameters), the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties.
|
||||
The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties.
|
||||
|
||||
From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained.
|
||||
|
||||
#+name: tab:test_apa_piezo_properties
|
||||
@ -1379,18 +1356,103 @@ ka = cE*A/L; % Stiffness of the two stacks [N/m]
|
||||
ga = d33*n*ka; % Actuator Constant [N/V]
|
||||
#+end_src
|
||||
|
||||
**** Identification of the APA Characteristics
|
||||
|
||||
Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Load reduced order model
|
||||
K = readmatrix('APA95ML_K.CSV'); % order: 48
|
||||
M = readmatrix('APA95ML_M.CSV');
|
||||
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
|
||||
#+end_src
|
||||
|
||||
The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure ref:fig:detail_fem_apa95ml_compliance), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames.
|
||||
The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML.
|
||||
A value of $23\,N/\mu m$ was found which is close to the specified stiffness in the datasheet of $k = 21\,N/\mu m$.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Stiffness estimation
|
||||
m = 0.0001; % block-free condition, no payload
|
||||
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
|
||||
% The inverse of the DC gain of the transfer function
|
||||
% from vertical force to vertical displacement is the axial stiffness of the APA
|
||||
k_est = 1/dcgain(G); % [N/m]
|
||||
#+end_src
|
||||
|
||||
The multi-body model predicted a resonant frequency under block-free conditions of $2024\,\text{Hz}$ (Figure ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification of $2000\,\text{Hz}$.
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Estimated compliance of the APA95ML
|
||||
freqs = logspace(2, log10(5000), 1000);
|
||||
|
||||
% Get first resonance indice
|
||||
i_max = find(abs(squeeze(freqresp(G, freqs(2:end), 'Hz'))) - abs(squeeze(freqresp(G, freqs(1:end-1), 'Hz'))) < 0, 1);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'DisplayName', 'Compliance');
|
||||
plot([freqs(1), freqs(end)], [1/k_est, 1/k_est], 'k--', 'DisplayName', sprintf('$1/k$ ($k = %.0f N/\\mu m$)', 1e-6*k_est))
|
||||
xline(freqs(i_max), '--', 'linewidth', 1, 'color', [0,0,0], 'DisplayName', sprintf('$f_0 = %.0f$ Hz', freqs(i_max)))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([100, 5000]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_fem_apa95ml_compliance.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:detail_fem_apa95ml_compliance
|
||||
#+caption: Estimated compliance of the APA95ML
|
||||
#+RESULTS:
|
||||
[[file:figs/detail_fem_apa95ml_compliance.png]]
|
||||
|
||||
In order to estimate the stroke of the APA95ML, first the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, needs to be determined.
|
||||
This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of $1.5$ was derived.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Estimation of the amplification factor and Stroke
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
|
||||
% Estimated amplification factor
|
||||
ampl_factor = abs(dcgain(G(1,1))./dcgain(G(2,1)));
|
||||
|
||||
% Estimated stroke
|
||||
apa_stroke = ampl_factor * 3 * 20e-6; % [m]
|
||||
#+end_src
|
||||
|
||||
The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,mm$), produce an individual nominal stroke of $20\,\mu m$ (see data-sheet of the piezoelectric stacks on Table ref:tab:detail_fem_stack_parameters, page pageref:tab:detail_fem_stack_parameters).
|
||||
As three stacks are used, the horizontal displacement is $60\,\mu m$.
|
||||
Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\mu m$ which falls within the manufacturer-specified range of $80\,\mu m$ and $120\,\mu m$.
|
||||
|
||||
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model.
|
||||
|
||||
** Experimental Validation
|
||||
<<ssec:detail_fem_super_element_validation>>
|
||||
**** Introduction :ignore:
|
||||
**** Test Bench
|
||||
|
||||
goal: validation of the procedure.
|
||||
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
|
||||
The goal is to measure the dynamics of the APA95ML and compared it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
|
||||
|
||||
- Explain test bench: (Figure ref:fig:detail_fem_apa95ml_bench)
|
||||
- 5.7kg granite, vertical guided with an air bearing
|
||||
- fibered interferometer measured the vertical motion of the granite $y$
|
||||
- DAC generating control signal $u$, voltage amplifier gain of 20, $V_a$ is the voltage across the two piezoelectric stacks
|
||||
- ADC is used to measured the voltage across the piezoelectric sensor stack
|
||||
The test bench illustrated in Figure ref:fig:detail_fem_apa95ml_bench was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML.
|
||||
The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$.
|
||||
A digital-to-analog converter (DAC) was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplification stage providing a gain factor of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks.
|
||||
Measurement of the sensor stack voltage $V_s$ was performed using an analog-to-digital converter (ADC).
|
||||
|
||||
#+name: fig:detail_fem_apa95ml_bench
|
||||
#+caption: Test bench used to validate "reduced order solid bodies" using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}).
|
||||
@ -1412,15 +1474,21 @@ goal: validation of the procedure.
|
||||
|
||||
**** Comparison of the dynamics
|
||||
|
||||
- Explain how to experimentally measure the transfer function:
|
||||
- test signal, here noise
|
||||
- compute and show the transfer functions from $V_a$ to $y$ and to $V_s$
|
||||
- Compare the model and measurement: validation (Figure ref:fig:detail_fem_apa95ml_comp_plant)
|
||||
- talk about the phase:
|
||||
- for force sensor, just delay linked to the limited sampling rate of $0.1\,ms$
|
||||
- for interferometer: additional delay due to electronics being used
|
||||
- good match. The gains can be further tuned based on the experimental results.
|
||||
- [ ] talk about minimum phase zero: will be discussed during the experimental phase
|
||||
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
|
||||
The identification procedure necessitated careful choice of the excitation signal [[cite:&pintelon12_system_ident, chap. 5]].
|
||||
The most used ones are impulses (particularly suited to modal analysis), steps, random noise signals, and multi-sine excitations.
|
||||
During all this experimental work, random noise excitation was predominantly employed.
|
||||
|
||||
The designed excitation signal is then generated and both input and output signals are synchronously acquired.
|
||||
From the obtained input and output data, the frequency response functions were derived.
|
||||
To improve the quality of the obtained frequency domain data, averaging and windowing were used [[cite:&pintelon12_system_ident, chap. 13]]..
|
||||
|
||||
The obtained frequency response functions from $V_a$ to $V_s$ and to $y$ are compared with the theoretical predictions derived from the multi-body model in Figure ref:fig:detail_fem_apa95ml_comp_plant.
|
||||
|
||||
The difference in phase between the model and the measurements can be attributed to the sampling time of $0.1\,ms$ and to additional delays induced by electronic instrumentation related to the interferometer.
|
||||
The presence of a non-minimum phase zero in the measured system response (Figure ref:fig:detail_fem_apa95ml_comp_plant_sensor), shall be addressed during the experimental phase.
|
||||
|
||||
Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be further refined through calibration against the experimental data.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Experimental plant identification
|
||||
@ -1530,10 +1598,10 @@ exportFig('figs/detail_fem_apa95ml_comp_plant_sensor.pdf', 'width', 'half', 'hei
|
||||
#+end_src
|
||||
|
||||
#+name: fig:detail_fem_apa95ml_comp_plant
|
||||
#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})
|
||||
#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $y$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $d_i$}
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
|
||||
#+attr_latex: :options {0.49\textwidth}
|
||||
#+begin_subfigure
|
||||
#+attr_latex: :width 0.95\linewidth
|
||||
@ -1549,20 +1617,20 @@ exportFig('figs/detail_fem_apa95ml_comp_plant_sensor.pdf', 'width', 'half', 'hei
|
||||
|
||||
**** Integral Force Feedback with APA
|
||||
|
||||
goal:
|
||||
- validate the use of super element for control tasks
|
||||
To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally.
|
||||
Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains.
|
||||
|
||||
The controller used in the Integral Force Feedback Architecture is eqref:eq:detail_fem_iff_controller, wtih $g$ a gain that can be tuned.
|
||||
The IFF controller implementation, defined in equation ref:eq:detail_fem_iff_controller, incorporated a tunable gain parameter $g$ and was designed to provide integral action near the system resonances and to limit the low frequency gain using an high pass filter.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_iff_controller}
|
||||
K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
|
||||
\end{equation}
|
||||
|
||||
The theoretical damped dynamics of the closed-loop system was analyzed through using the model by computed the root locus plot shown in Figure ref:fig:detail_fem_apa95ml_iff_root_locus.
|
||||
For experimental validation, six gain values were tested: $g = [0,\,10,\,50,\,100,\,500,\,1000]$.
|
||||
The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure ref:fig:detail_fem_apa95ml_damped_plants.
|
||||
|
||||
Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.
|
||||
In the frequency band of interest, this controller should mostly act as a pure integrator.
|
||||
|
||||
- [ ] Maybe make a block diagram of the control with added damped input
|
||||
The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics, thereby validating its utility for control system design and analysis.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Integral Force Feedback Controller
|
||||
@ -1696,7 +1764,7 @@ exportFig('figs/detail_fem_apa95ml_damped_plants.pdf', 'width', 'half', 'height'
|
||||
#+end_src
|
||||
|
||||
#+name: fig:detail_fem_apa95ml_iff_results
|
||||
#+caption: Obtained results using Integral Force Feedback with the APA95ML.
|
||||
#+caption: Obtained results using Integral Force Feedback with the APA95ML. Obtained closed-loop poles as a function of the controller gain $g$ are prediction by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.
|
||||
#+attr_latex: :options [htbp]
|
||||
#+begin_figure
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
|
||||
@ -1718,11 +1786,13 @@ exportFig('figs/detail_fem_apa95ml_damped_plants.pdf', 'width', 'half', 'height'
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
|
||||
- Validation of the method
|
||||
- Very useful to optimize different parts
|
||||
- However, model order may become very large and not convenient to perform time domain simulations
|
||||
- But extracting dynamics is not computational intensive, even for large model orders
|
||||
- For instance APA: order 48, 6 APA for the nano hexapod 288 orders just for the APA
|
||||
The modeling procedure presented in this section will demonstrate significant utility for the optimization of complex mechanical components within multi-body systems, particularly in the design of actuators (Section ref:sec:detail_fem_actuator) and flexible joints (Section ref:sec:detail_fem_joint).
|
||||
|
||||
Through experimental validation using an Amplified Piezoelectric Actuator, the methodology has been shown to accurately predict both open-loop and closed-loop dynamic behavior, thereby establishing its reliability for component design and system analysis.
|
||||
|
||||
While this modeling approach provides accurate predictions of component behavior, the resulting model order can become prohibitively high for practical time-domain simulations.
|
||||
This is exemplified by the nano-hexapod configuration, where the implementation of six Amplified Piezoelectric Actuators, each modeled with 48 degrees of freedom, yields 288 degrees of freedom only for the actuators.
|
||||
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
|
||||
|
||||
* Actuator
|
||||
<<sec:detail_fem_actuator>>
|
||||
@ -1895,104 +1965,6 @@ To validate the choice of the APA300ML (Shown in Figure ref:fig:detail_fem_apa30
|
||||
- The link between mechanical properties and electrical properties was discussed in Section ref:ssec:detail_fem_super_element_validation.
|
||||
As the stacks are the same between the APA300ML and the APA95ML, the values estimated for $g_a$ and $g_s$ are used for the APA300ML.
|
||||
|
||||
** Identification of the APA Characteristics
|
||||
**** Introduction :ignore:
|
||||
A first validation of the FEM and inclusion of the "reduced order flexible model" in the multi body-model is performed by computed some key characteristics of the APA that can be compared against the datasheet.
|
||||
|
||||
#+begin_src matlab
|
||||
% Extract the stiffness and mass matrices
|
||||
K = readmatrix('APA300ML_mat_K.CSV');
|
||||
M = readmatrix('APA300ML_mat_M.CSV');
|
||||
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_out_nodes_3D.txt');
|
||||
#+end_src
|
||||
|
||||
**** Stiffness
|
||||
The stiffness is estimated by extracting the transfer function from a vertical force applied on the top frame to the displacement of the same top frame.
|
||||
The inverse of the DC gain this transfer function should be equal to the axial stiffness of the APA300ML.
|
||||
A value of $1.75\,N/\mu m$ is found which is close to the specified stiffness in the datasheet of $k = 1.8\,N/\mu m$.
|
||||
See compliance transfer function ref:fig:detail_fem_apa300ml_compliance.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Stiffness estimation
|
||||
m = 0.0001; % block-free condition, no payload
|
||||
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
|
||||
% The inverse of the DC gain of the transfer function
|
||||
% from vertical force to vertical displacement is the axial stiffness of the APA
|
||||
k_est = 1/dcgain(G); % [N/m]
|
||||
#+end_src
|
||||
|
||||
|
||||
**** Resonance Frequency
|
||||
|
||||
The resonance frequency in the block-free condition is specified to be between 650Hz and 840Hz.
|
||||
This is estimated at 709Hz from the model (Figure ref:fig:detail_fem_apa300ml_compliance).
|
||||
|
||||
#+begin_src matlab :exports none :results none
|
||||
%% Estimated compliance of the APA300ML
|
||||
freqs = logspace(2, log10(5000), 1000);
|
||||
|
||||
% Get first resonance indice
|
||||
i_max = find(abs(squeeze(freqresp(G, freqs(2:end), 'Hz'))) - abs(squeeze(freqresp(G, freqs(1:end-1), 'Hz'))) < 0, 1);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'DisplayName', 'Compliance');
|
||||
plot([freqs(1), freqs(end)], [1/k_est, 1/k_est], 'k--', 'DisplayName', sprintf('$1/k$ ($k = %.2f N/\\mu m$)', 1e-6*k_est))
|
||||
xline(freqs(i_max), '--', 'linewidth', 1, 'color', [0,0,0], 'DisplayName', sprintf('$f_0 = %.0f$ Hz', freqs(i_max)))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||||
leg.ItemTokenSize(1) = 15;
|
||||
xlim([100, 5000]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/detail_fem_apa300ml_compliance.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:detail_fem_apa300ml_compliance
|
||||
#+caption: Estimated compliance of the APA300ML
|
||||
#+RESULTS:
|
||||
[[file:figs/detail_fem_apa300ml_compliance.png]]
|
||||
|
||||
|
||||
**** Amplification Factor and Actuator Stroke
|
||||
|
||||
The amplification factor is the ratio of the vertical displacement to the (horizontal) stack displacement.
|
||||
It can be estimated from the multi-body model by computing the transfer function from the horizontal motion of the stacks to the vertical motion of the APA.
|
||||
The ratio between the two is found to be equal to $5$.
|
||||
This is linked to the
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Estimation of the amplification factor and Stroke
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
|
||||
% Estimated amplification factor
|
||||
ampl_factor = abs(dcgain(G(1,1))./dcgain(G(2,1)));
|
||||
|
||||
% Estimated stroke
|
||||
apa_stroke = ampl_factor * 3 * 20e-6; % [m]
|
||||
#+end_src
|
||||
|
||||
From the data-sheet of the piezoelectric stacks (see Table ref:tab:detail_fem_stack_parameters, page pageref:tab:detail_fem_stack_parameters), the nominal stroke of the stack is $20\,\mu m$ (which is typical for PZT to have a maximum stroke equal to $0.1\,\%$ of its length, here equal to $20\,mm$).
|
||||
Three stacks are used, for an horizontal stroke of the stacks of $60\,\mu m$.
|
||||
With an amplification factor equal to $5$, the vertical stroke is estimated at $300\,\mu m$, which corresponds to what is indicated in the datasheet.
|
||||
|
||||
|
||||
This analysis provides some confidence on the model accuracy.
|
||||
|
||||
** Simpler 2DoF Model of the APA300ML
|
||||
<<sec:apa_model>>
|
||||
**** Introduction :ignore:
|
||||
|
||||
BIN
nass-fem.pdf
BIN
nass-fem.pdf
Binary file not shown.
409
nass-fem.tex
409
nass-fem.tex
@ -1,4 +1,4 @@
|
||||
% Created 2025-02-26 Wed 09:37
|
||||
% Created 2025-02-26 Wed 15:42
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
|
||||
|
||||
@ -43,71 +43,51 @@ To do so, Reduced Order Flexible Bodies are used (Section \ref{sec:detail_fem_su
|
||||
\end{itemize}
|
||||
|
||||
\chapter{Reduced order flexible bodies}
|
||||
\label{sec:orgefd3374}
|
||||
\label{sec:org5704c94}
|
||||
\label{sec:detail_fem_super_element}
|
||||
Goal:
|
||||
\begin{itemize}
|
||||
\item include parts from which dynamical properties are estimated from a FEM
|
||||
\end{itemize}
|
||||
Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
|
||||
These components are traditionally analyzed using Finite Element Analysis (FEA) software.
|
||||
However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models \cite{hatch00_vibrat_matlab_ansys}.
|
||||
This combined multibody-FEA modeling approach presents significant advantages, as it enables the selective application of FEA modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system \cite{rankers98_machin}.
|
||||
|
||||
Outline:
|
||||
\begin{itemize}
|
||||
\item Quick explanation of the theory
|
||||
\item Explain the implementation with FEA software (Ansys) and Simscape
|
||||
\item Experimental validation with an amplified piezoelectric actuator
|
||||
\end{itemize}
|
||||
|
||||
\cite{rankers98_machin}
|
||||
\cite{hatch00_vibrat_matlab_ansys}
|
||||
\section{FEA Modal Reduction}
|
||||
\label{sec:org4844a44}
|
||||
The investigation of this hybrid modeling approach is structured in three sections.
|
||||
First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section \ref{ssec:detail_fem_super_element_theory}).
|
||||
It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section \ref{ssec:detail_fem_super_element_example}).
|
||||
Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section \ref{ssec:detail_fem_super_element_validation}).
|
||||
\section{Procedure}
|
||||
\label{sec:orga74dca6}
|
||||
\label{ssec:detail_fem_super_element_theory}
|
||||
|
||||
\begin{itemize}
|
||||
\item sub-components in the multi-body model as reduced order flexible bodies representing the component's modal behaviour with reduced mass and stiffness matrices obtained from finite element analysis (FEA) models
|
||||
\item matrices were created from FEA models via modal reduction techniques, more specifically the component mode synthesis (CMS).
|
||||
\item this makes this design approach a combined multibody-FEA technique.
|
||||
\end{itemize}
|
||||
In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models.
|
||||
These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis (CMS), thus establishing this design approach as a combined multibody-FEA methodology.
|
||||
|
||||
Standard FEA implementations typically involve thousands or even hundreds of thousands of DoF, rendering direct integration into multi-body simulations computationally prohibitive.
|
||||
The objective of modal reduction is therefore to substantially decrease the number of DoF while preserving the essential dynamic characteristics of the component.
|
||||
|
||||
\begin{itemize}
|
||||
\item FEM: high number of DoF
|
||||
\item goal: reduce number of DoF, allow to integrate in multi-body simulation
|
||||
\end{itemize}
|
||||
The procedure for implementing this reduction involves several distinct stages.
|
||||
Initially, the component is modeled in a finite element software with appropriate material properties and boundary conditions.
|
||||
Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component.
|
||||
These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames.
|
||||
|
||||
Procedure:
|
||||
\begin{itemize}
|
||||
\item model the part in FE software as usually by defining material properties, etc.
|
||||
\item define frames for which we want to the multi-body model will then be able to interface with, and can be used to:
|
||||
\begin{itemize}
|
||||
\item connect other parts
|
||||
\item apply forces and torques
|
||||
\item measure motion between frames
|
||||
\end{itemize}
|
||||
\item perform the modal reduction technique from FEA (also called component mode synthesis or ``Craig-Bampton'' method \cite{craig68_coupl_subst_dynam_analy}) for the reduction of the high number of FEA degrees of freedom (DoF) to a smaller number of retained degrees of freedom
|
||||
typically from hundred thousands to less than 100 DoF
|
||||
\item the number of DoF is 6 times the number of defined frame + any number of additional DoF that we want to model
|
||||
\(m = 6 \times n + p\)
|
||||
\(n\) the number of frames, \(p\) the number of additional modes
|
||||
\item then, it outputs \(m \times m\) reduced mass and stiffness matrices
|
||||
\item in the multi-body model, the two reduced matrices can be used to model the part
|
||||
\end{itemize}
|
||||
Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method \cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that transforms the extensive FEA degrees of freedom into a significantly reduced set of retained degrees of freedom.
|
||||
This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF.
|
||||
The number of degrees of freedom in the reduced model is determined by \eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled.
|
||||
The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior.
|
||||
|
||||
\section{Validation of the Method}
|
||||
\label{sec:orgcb6472b}
|
||||
\label{ssec:detail_fem_super_element_validation}
|
||||
Validation with Amplified Piezoelectric Actuator, because:
|
||||
\begin{itemize}
|
||||
\item is a good candidate for the nano-hexapod (as will be explained in Section \ref{sec:detail_fem_actuator})
|
||||
\item had one in the lab for experimental testing (APA95ML, Figure \ref{fig:detail_fem_apa95ml_picture})
|
||||
It is composed of several piezoelectric stacks (arranged horizontally, in blue), and a shell (in red) that amplifies the motion. The working direction of the APA95ML is vertical.
|
||||
\item permits to model a mechanical structure (similar to a flexible joint), piezoelectric actuator and piezoelectric sensor
|
||||
\end{itemize}
|
||||
\begin{equation}\label{eq:detail_fem_model_order}
|
||||
m = 6 \times n + p
|
||||
\end{equation}
|
||||
|
||||
Quick explanation of APA:
|
||||
\begin{itemize}
|
||||
\item \cite{claeyssen07_amplif_piezoel_actuat}
|
||||
\end{itemize}
|
||||
\section{Example with an Amplified Piezoelectric Actuator}
|
||||
\label{sec:org3e7c2ec}
|
||||
\label{ssec:detail_fem_super_element_example}
|
||||
The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons.
|
||||
Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section \ref{sec:detail_fem_actuator}.
|
||||
Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure \ref{fig:detail_fem_apa95ml_picture}) was available in the laboratory for experimental testing.
|
||||
|
||||
The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure \ref{fig:detail_fem_apa95ml_picture}) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement into the vertical direction \cite{claeyssen07_amplif_piezoel_actuat}.
|
||||
The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework.
|
||||
The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation.
|
||||
|
||||
\begin{minipage}[b]{0.48\linewidth}
|
||||
\begin{center}
|
||||
@ -118,40 +98,22 @@ Quick explanation of APA:
|
||||
\hfill
|
||||
\begin{minipage}[b]{0.48\linewidth}
|
||||
\centering
|
||||
\begin{tabularx}{0.8\linewidth}{Xcc}
|
||||
\begin{tabularx}{0.7\linewidth}{Xc}
|
||||
\toprule
|
||||
Parameter & Unit & Value\\
|
||||
\textbf{Parameter} & \textbf{Value}\\
|
||||
\midrule
|
||||
Nominal Stroke & \(\mu m\) & 100\\
|
||||
Blocked force & \(N\) & 1600\\
|
||||
Stiffness & \(N/\mu m\) & 16\\
|
||||
Nominal Stroke & \(100\,\mu m\)\\
|
||||
Blocked force & \(2100\,N\)\\
|
||||
Stiffness & \(21\,N/\mu m\)\\
|
||||
\bottomrule
|
||||
\end{tabularx}
|
||||
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
|
||||
\end{minipage}
|
||||
\paragraph{Finite Element Model}
|
||||
\label{sec:orgf976215}
|
||||
\label{sec:org491eeae}
|
||||
|
||||
\begin{itemize}
|
||||
\item explain how the FEM is done:
|
||||
\begin{itemize}
|
||||
\item material properties (Table \ref{tab:detail_fem_material_properties})
|
||||
\item mesh (Figure \ref{fig:detail_fem_apa95ml_mesh})
|
||||
\end{itemize}
|
||||
\item explain piezoelectric materials:
|
||||
\begin{itemize}
|
||||
\item sensors
|
||||
\item actuators
|
||||
\end{itemize}
|
||||
\item choice of frames (Figure \ref{fig:detail_fem_apa95ml_frames})
|
||||
\begin{itemize}
|
||||
\item 2 for each piezoelectric stack to measure strain and apply forces
|
||||
\item 1 at the top, 1 at the bottom to connect to other elements
|
||||
\end{itemize}
|
||||
\item choose number of DoF => size of model
|
||||
7 frames + 6 modes => order 48
|
||||
\item perform the reduction: show the output reduced matrices
|
||||
\end{itemize}
|
||||
The development of the finite element model for the APA95ML necessitated the specification of appropriate material properties, as summarized in Table \ref{tab:detail_fem_material_properties}.
|
||||
The finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated.
|
||||
|
||||
\begin{table}[htbp]
|
||||
\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA modal reduction model. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density}
|
||||
@ -166,67 +128,59 @@ Piezoelectric Ceramics (PZT) & \(49.5\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
The definition of interface frames, or ``remote points'', constitute a critical aspect of the model preparation.
|
||||
Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation.
|
||||
|
||||
Six additional modes were considered, resulting in total model order of \(48\).
|
||||
The modal reduction procedure was then executed, yielding the reduced mass and stiffness matrices that form the foundation of the component's representation in the multi-body simulation environment.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_mesh.png}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_mesh.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh and "remote points"}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.85\linewidth]{figs/detail_fem_apa95ml_frames.png}
|
||||
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa_modal_schematic.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_frames}Defined frames}
|
||||
\subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_fem_apa95ml_model}Finite element model of the APA95ML. Obtained mesh is shown in (\subref{fig:detail_fem_apa95ml_mesh}). Frames (or ``remote points'') used for the modal reduction are shown in (\subref{fig:detail_fem_apa95ml_frames}).}
|
||||
\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_modal_schematic}).}
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Super Element in the Multi-Body Model}
|
||||
\label{sec:orga1214e3}
|
||||
\label{sec:org29dd028}
|
||||
|
||||
Model:
|
||||
\begin{itemize}
|
||||
\item Connect frame \(\{4\}\) to world frame and frame \(\{6\}\) to a 5.5kg mass, vertically guided
|
||||
\item 2 actuator stacks, 1 sensor stack:
|
||||
\begin{itemize}
|
||||
\item force source between frames \(\{3\}\) and \(\{2\}\)
|
||||
\item measured strain for force sensor by measuring the displacement between \(\{1\}\) and \(\{7\}\)
|
||||
\end{itemize}
|
||||
\item Input: internal force applied
|
||||
\item Output: strain in the sensor stack
|
||||
\item Issue: how to convert voltage to force and strain to voltage?
|
||||
\end{itemize}
|
||||
Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called ``Reduced Order Flexible Solid''.
|
||||
This block has several interface frames corresponding to the ones defined in the FEA software.
|
||||
Frame \(\{4\}\) was connected to the ``world'' frame, while frame \(\{6\}\) was coupled to a vertically guided payload.
|
||||
In this example, two piezoelectric stacks were used for actuation while one piezoelectric stack was used as a force sensor.
|
||||
Therefore, a force source \(F_a\) operating between frames \(\{3\}\) and \(\{2\}\) was used, while a displacement sensor \(d_L\) between frames \(\{1\}\) and \(\{7\}\) was used for the sensor stack.
|
||||
This is illustrated in Figure \ref{fig:detail_fem_apa_model_schematic}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_fem_apa_modal_schematic.png}
|
||||
\caption{\label{fig:detail_fem_apa_model_schematic}Amplified Piezoelectric Actuator Schematic}
|
||||
\end{figure}
|
||||
|
||||
Need to link the electrical domain (voltages, charges) with the mechanical domain (forces, strain).
|
||||
To do so, ``actuator constant'' \(g_a\) and ``sensor constant'' \(g_s\) are used as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.
|
||||
|
||||
A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):
|
||||
\begin{equation}
|
||||
\boxed{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}
|
||||
\boxed{V_s = g_s \cdot dl}
|
||||
\end{equation}
|
||||
|
||||
In order to correctly model the piezoelectric actuator with Simscape, the values for \(g_a\) and \(g_s\) needs to be determined.
|
||||
\begin{itemize}
|
||||
\item \(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack
|
||||
\item \(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\)
|
||||
\end{itemize}
|
||||
However, to have access to the physical voltage input of the actuators stacks \(V_a\) and to the generated voltage by the force sensor \(V_s\), conversion between the electrical and mechanical domains need to be determined.
|
||||
|
||||
\paragraph{Sensor and Actuator ``constants''}
|
||||
\label{sec:orgb6b6d3f}
|
||||
\label{sec:org1329f1a}
|
||||
|
||||
The gains \(g_a\) and \(g_s\) were estimated from the physical properties of the piezoelectric stack material (summarized in Table \ref{tab:detail_fem_stack_parameters}).
|
||||
To link the electrical domain to the mechanical domain, an ``actuator constant'' \(g_a\) and a ``sensor constant'' \(g_s\) were introduced as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.
|
||||
|
||||
From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:detail_fem_dl_to_vs}.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_dl_to_vs}
|
||||
V_s = g_s \cdot d_L, \quad g_s = \frac{d_{33}}{\epsilon^T s^D n}
|
||||
\end{equation}
|
||||
|
||||
From \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:detail_fem_va_to_fa}.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_va_to_fa}
|
||||
F_a = g_a \cdot V_a, \quad g_a = d_{33} n k_a, \quad k_a = \frac{c^{E} A}{L}
|
||||
\end{equation}
|
||||
|
||||
Unfortunately, it is difficult to know exactly which material is used in the amplified piezoelectric actuator\footnote{The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.}.
|
||||
However, based on the available properties of the stacks in the data-sheet (summarized in Table \ref{tab:detail_fem_stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
|
||||
|
||||
\begin{table}[htbp]
|
||||
\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters}
|
||||
@ -246,19 +200,7 @@ Stack Area & \(mm^2\) & 10x10\\
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}.
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
|
||||
F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Unfortunately, it is difficult to know exactly which material is used in the amplified piezoelectric actuator\footnote{The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.}.
|
||||
However, based on the available properties of the stacks in the data-sheet (summarized in Table \ref{tab:detail_fem_stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
|
||||
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
|
||||
|
||||
From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained.
|
||||
|
||||
\begin{table}[htbp]
|
||||
@ -279,20 +221,42 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
\paragraph{Experimental Validation}
|
||||
\label{sec:orgfd4d8f6}
|
||||
\paragraph{Identification of the APA Characteristics}
|
||||
\label{sec:org5512e6c}
|
||||
|
||||
goal: validation of the procedure.
|
||||
Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications.
|
||||
|
||||
\begin{itemize}
|
||||
\item Explain test bench: (Figure \ref{fig:detail_fem_apa95ml_bench})
|
||||
\begin{itemize}
|
||||
\item 5.7kg granite, vertical guided with an air bearing
|
||||
\item fibered interferometer measured the vertical motion of the granite \(y\)
|
||||
\item DAC generating control signal \(u\), voltage amplifier gain of 20, \(V_a\) is the voltage across the two piezoelectric stacks
|
||||
\item ADC is used to measured the voltage across the piezoelectric sensor stack
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure \ref{fig:detail_fem_apa95ml_compliance}), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames.
|
||||
The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML.
|
||||
A value of \(23\,N/\mu m\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,N/\mu m\).
|
||||
|
||||
The multi-body model predicted a resonant frequency under block-free conditions of \(2024\,\text{Hz}\) (Figure \ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification of \(2000\,\text{Hz}\).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/detail_fem_apa95ml_compliance.png}
|
||||
\caption{\label{fig:detail_fem_apa95ml_compliance}Estimated compliance of the APA95ML}
|
||||
\end{figure}
|
||||
|
||||
In order to estimate the stroke of the APA95ML, first the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, needs to be determined.
|
||||
This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of \(1.5\) was derived.
|
||||
|
||||
The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,mm\)), produce an individual nominal stroke of \(20\,\mu m\) (see data-sheet of the piezoelectric stacks on Table \ref{tab:detail_fem_stack_parameters}, page \pageref{tab:detail_fem_stack_parameters}).
|
||||
As three stacks are used, the horizontal displacement is \(60\,\mu m\).
|
||||
Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu m\) which falls within the manufacturer-specified range of \(80\,\mu m\) and \(120\,\mu m\).
|
||||
|
||||
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model.
|
||||
|
||||
\section{Experimental Validation}
|
||||
\label{sec:org8627abc}
|
||||
\label{ssec:detail_fem_super_element_validation}
|
||||
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
|
||||
The goal is to measure the dynamics of the APA95ML and compared it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
|
||||
|
||||
The test bench illustrated in Figure \ref{fig:detail_fem_apa95ml_bench} was used, which consists of a \(5.7\,kg\) granite suspended on top of the APA95ML.
|
||||
The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\).
|
||||
A digital-to-analog converter (DAC) was used to generate the control signal \(u\), which was subsequently conditioned through a voltage amplification stage providing a gain factor of \(20\), ultimately yielding the effective voltage \(V_a\) across the two piezoelectric stacks.
|
||||
Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to-digital converter (ADC).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.34\textwidth}
|
||||
@ -309,29 +273,31 @@ goal: validation of the procedure.
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_fem_apa95ml_bench}Test bench used to validate ``reduced order solid bodies'' using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}).}
|
||||
\end{figure}
|
||||
\paragraph{Comparison of the dynamics}
|
||||
\label{sec:orgb3fa207}
|
||||
|
||||
\begin{itemize}
|
||||
\item Explain how to experimentally measure the transfer function:
|
||||
\begin{itemize}
|
||||
\item test signal, here noise
|
||||
\item compute and show the transfer functions from \(V_a\) to \(y\) and to \(V_s\)
|
||||
\item Compare the model and measurement: validation (Figure \ref{fig:detail_fem_apa95ml_comp_plant})
|
||||
\item talk about the phase:
|
||||
\begin{itemize}
|
||||
\item for force sensor, just delay linked to the limited sampling rate of \(0.1\,ms\)
|
||||
\item for interferometer: additional delay due to electronics being used
|
||||
\end{itemize}
|
||||
\item good match. The gains can be further tuned based on the experimental results.
|
||||
\item[{$\square$}] talk about minimum phase zero: will be discussed during the experimental phase
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
|
||||
The identification procedure necessitated careful choice of the excitation signal \cite[, chap. 5]{pintelon12_system_ident}.
|
||||
The most used ones are impulses (particularly suited to modal analysis), steps, random noise signals, and multi-sine excitations.
|
||||
During all this experimental work, random noise excitation was predominantly employed.
|
||||
|
||||
The designed excitation signal is then generated and both input and output signals are synchronously acquired.
|
||||
From the obtained input and output data, the frequency response functions were derived.
|
||||
To improve the quality of the obtained frequency domain data, averaging and windowing were used \cite[, chap. 13]{pintelon12_system_ident}..
|
||||
|
||||
The obtained frequency response functions from \(V_a\) to \(V_s\) and to \(y\) are compared with the theoretical predictions derived from the multi-body model in Figure \ref{fig:detail_fem_apa95ml_comp_plant}.
|
||||
|
||||
The difference in phase between the model and the measurements can be attributed to the sampling time of \(0.1\,ms\) and to additional delays induced by electronic instrumentation related to the interferometer.
|
||||
The presence of a non-minimum phase zero in the measured system response (Figure \ref{fig:detail_fem_apa95ml_comp_plant_sensor}), shall be addressed during the experimental phase.
|
||||
|
||||
Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could be further refined through calibration against the experimental data.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_comp_plant_actuator.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $d_i$}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
\begin{center}
|
||||
@ -339,30 +305,26 @@ goal: validation of the procedure.
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(d_i\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})}
|
||||
\caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(y\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})}
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Integral Force Feedback with APA}
|
||||
\label{sec:orgbd44486}
|
||||
\label{sec:org182828d}
|
||||
|
||||
goal:
|
||||
\begin{itemize}
|
||||
\item validate the use of super element for control tasks
|
||||
\end{itemize}
|
||||
To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally.
|
||||
Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains.
|
||||
|
||||
The controller used in the Integral Force Feedback Architecture is \eqref{eq:detail_fem_iff_controller}, wtih \(g\) a gain that can be tuned.
|
||||
The IFF controller implementation, defined in equation \ref{eq:detail_fem_iff_controller}, incorporated a tunable gain parameter \(g\) and was designed to provide integral action near the system resonances and to limit the low frequency gain using an high pass filter.
|
||||
|
||||
\begin{equation}\label{eq:detail_fem_iff_controller}
|
||||
K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
|
||||
\end{equation}
|
||||
|
||||
The theoretical damped dynamics of the closed-loop system was analyzed through using the model by computed the root locus plot shown in Figure \ref{fig:detail_fem_apa95ml_iff_root_locus}.
|
||||
For experimental validation, six gain values were tested: \(g = [0,\,10,\,50,\,100,\,500,\,1000]\).
|
||||
The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure \ref{fig:detail_fem_apa95ml_damped_plants}.
|
||||
|
||||
Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.
|
||||
In the frequency band of interest, this controller should mostly act as a pure integrator.
|
||||
|
||||
\begin{itemize}
|
||||
\item[{$\square$}] Maybe make a block diagram of the control with added damped input
|
||||
\end{itemize}
|
||||
The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics, thereby validating its utility for control system design and analysis.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
@ -377,23 +339,21 @@ In the frequency band of interest, this controller should mostly act as a pure i
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_fem_apa95ml_iff_results}Obtained results using Integral Force Feedback with the APA95ML.}
|
||||
\caption{\label{fig:detail_fem_apa95ml_iff_results}Obtained results using Integral Force Feedback with the APA95ML. Obtained closed-loop poles as a function of the controller gain \(g\) are prediction by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.}
|
||||
\end{figure}
|
||||
|
||||
\section*{Conclusion}
|
||||
\label{sec:orga81cb67}
|
||||
\begin{itemize}
|
||||
\item Validation of the method
|
||||
\item Very useful to optimize different parts
|
||||
\item However, model order may become very large and not convenient to perform time domain simulations
|
||||
\item But extracting dynamics is not computational intensive, even for large model orders
|
||||
\item For instance APA: order 48, 6 APA for the nano hexapod 288 orders just for the APA
|
||||
\label{sec:org7af3b1c}
|
||||
The modeling procedure presented in this section will demonstrate significant utility for the optimization of complex mechanical components within multi-body systems, particularly in the design of actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}).
|
||||
|
||||
\item[{$\square$}] \href{file:///home/thomas/Cloud/research/papers/published/brumund21\_multib\_simul\_reduc\_order\_flexib\_bodies\_fea/paper/brumund21\_multib\_simul\_reduc\_order\_flexib\_bodies\_fea.pdf}{published paper}
|
||||
\end{itemize}
|
||||
Through experimental validation using an Amplified Piezoelectric Actuator, the methodology has been shown to accurately predict both open-loop and closed-loop dynamic behavior, thereby establishing its reliability for component design and system analysis.
|
||||
|
||||
While this modeling approach provides accurate predictions of component behavior, the resulting model order can become prohibitively high for practical time-domain simulations.
|
||||
This is exemplified by the nano-hexapod configuration, where the implementation of six Amplified Piezoelectric Actuators, each modeled with 48 degrees of freedom, yields 288 degrees of freedom only for the actuators.
|
||||
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
|
||||
|
||||
\chapter{Actuator}
|
||||
\label{sec:orgece4287}
|
||||
\label{sec:orgcb23435}
|
||||
\label{sec:detail_fem_actuator}
|
||||
Goals:
|
||||
\begin{itemize}
|
||||
@ -404,7 +364,7 @@ and validate this choice with simulations
|
||||
\item Development of a 2DoF model for lower order models (i.e. for simulations)
|
||||
\end{itemize}
|
||||
\section{Choice of the Actuator based on Specifications}
|
||||
\label{sec:org058dd07}
|
||||
\label{sec:org6a6861c}
|
||||
\label{ssec:detail_fem_actuator_specifications}
|
||||
|
||||
From previous analysis:
|
||||
@ -515,7 +475,7 @@ Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
|
||||
\end{table}
|
||||
|
||||
\section{APA300ML - Reduced Order Flexible Body}
|
||||
\label{sec:orgcc3207d}
|
||||
\label{sec:org56a3ff1}
|
||||
\label{ssec:detail_fem_actuator_apa300ml}
|
||||
|
||||
To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa300ml_picture}):
|
||||
@ -551,47 +511,8 @@ To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa3
|
||||
As the stacks are the same between the APA300ML and the APA95ML, the values estimated for \(g_a\) and \(g_s\) are used for the APA300ML.
|
||||
\end{itemize}
|
||||
|
||||
\section{Identification of the APA Characteristics}
|
||||
\label{sec:org0b219f1}
|
||||
A first validation of the FEM and inclusion of the ``reduced order flexible model'' in the multi body-model is performed by computed some key characteristics of the APA that can be compared against the datasheet.
|
||||
|
||||
\paragraph{Stiffness}
|
||||
\label{sec:orgebcd8db}
|
||||
The stiffness is estimated by extracting the transfer function from a vertical force applied on the top frame to the displacement of the same top frame.
|
||||
The inverse of the DC gain this transfer function should be equal to the axial stiffness of the APA300ML.
|
||||
A value of \(1.75\,N/\mu m\) is found which is close to the specified stiffness in the datasheet of \(k = 1.8\,N/\mu m\).
|
||||
See compliance transfer function \ref{fig:detail_fem_apa300ml_compliance}.
|
||||
|
||||
\paragraph{Resonance Frequency}
|
||||
\label{sec:org7704c52}
|
||||
|
||||
The resonance frequency in the block-free condition is specified to be between 650Hz and 840Hz.
|
||||
This is estimated at 709Hz from the model (Figure \ref{fig:detail_fem_apa300ml_compliance}).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/detail_fem_apa300ml_compliance.png}
|
||||
\caption{\label{fig:detail_fem_apa300ml_compliance}Estimated compliance of the APA300ML}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\paragraph{Amplification Factor and Actuator Stroke}
|
||||
\label{sec:org218b81c}
|
||||
|
||||
The amplification factor is the ratio of the vertical displacement to the (horizontal) stack displacement.
|
||||
It can be estimated from the multi-body model by computing the transfer function from the horizontal motion of the stacks to the vertical motion of the APA.
|
||||
The ratio between the two is found to be equal to \(5\).
|
||||
This is linked to the
|
||||
|
||||
From the data-sheet of the piezoelectric stacks (see Table \ref{tab:detail_fem_stack_parameters}, page \pageref{tab:detail_fem_stack_parameters}), the nominal stroke of the stack is \(20\,\mu m\) (which is typical for PZT to have a maximum stroke equal to \(0.1\,\%\) of its length, here equal to \(20\,mm\)).
|
||||
Three stacks are used, for an horizontal stroke of the stacks of \(60\,\mu m\).
|
||||
With an amplification factor equal to \(5\), the vertical stroke is estimated at \(300\,\mu m\), which corresponds to what is indicated in the datasheet.
|
||||
|
||||
|
||||
This analysis provides some confidence on the model accuracy.
|
||||
|
||||
\section{Simpler 2DoF Model of the APA300ML}
|
||||
\label{sec:org3340d21}
|
||||
\label{sec:orgfabc6b8}
|
||||
\label{sec:apa_model}
|
||||
\begin{itemize}
|
||||
\item \emph{super-element} order is quite large, and therefore not practical for simulations
|
||||
@ -608,7 +529,7 @@ This analysis provides some confidence on the model accuracy.
|
||||
\item Therefore this model can be useful for simulations as it contains a very limited number of states, but when more complex dynamics of the APA is to be modelled, a flexible model will be used.
|
||||
\end{itemize}
|
||||
\paragraph{2DoF Model}
|
||||
\label{sec:org5603255}
|
||||
\label{sec:org5962dd3}
|
||||
|
||||
The model is adapted from \cite{souleille18_concep_activ_mount_space_applic}.
|
||||
|
||||
@ -638,7 +559,7 @@ The main advantage is that this model is very simple, only adds 4 states
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Parameter Tuning}
|
||||
\label{sec:org6d0757e}
|
||||
\label{sec:org7bd1971}
|
||||
9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:detail_fem_apa_2dof_model}) well represents the identified dynamics using the FEM.
|
||||
|
||||
\begin{itemize}
|
||||
@ -694,7 +615,7 @@ Of course, higher order modes are not represented by the 2DoF model, nor the lim
|
||||
\end{figure}
|
||||
|
||||
\section{Electrical characteristics of the APA}
|
||||
\label{sec:org02c143e}
|
||||
\label{sec:org5cdd335}
|
||||
|
||||
\begin{itemize}
|
||||
\item Mechanical equations and electrical equations are coupled
|
||||
@ -717,7 +638,7 @@ This will be discussed in chapter ``instrumentation''
|
||||
\end{figure}
|
||||
|
||||
\section{Validation with the Nano-Hexapod}
|
||||
\label{sec:org461915d}
|
||||
\label{sec:orgf89bb46}
|
||||
NASS model + FEM model (or just 2DoF) of APA300ML => validation (based on what?)
|
||||
|
||||
\begin{itemize}
|
||||
@ -755,7 +676,7 @@ here matrices have a size of 36
|
||||
|
||||
|
||||
\chapter{Flexible Joint}
|
||||
\label{sec:orgfd42b09}
|
||||
\label{sec:org8601117}
|
||||
\label{sec:detail_fem_joint}
|
||||
The flexible joints have few advantages compared to conventional joints such as the \textbf{absence of wear, friction and backlash} which allows extremely high-precision (predictable) motion.
|
||||
The parasitic bending and torsional stiffness of these joints usually induce some \textbf{limitation on the control performance}. \cite{mcinroy02_model_desig_flexur_joint_stewar}
|
||||
@ -784,7 +705,7 @@ Say that for simplicity (reduced number of parts, etc.), we consider the same jo
|
||||
\item Implementation of flexible elements in the Simscape model: close to simplified model
|
||||
\end{itemize}
|
||||
\section{Flexible joints for Stewart platforms}
|
||||
\label{sec:org5c169eb}
|
||||
\label{sec:orgd5923ee}
|
||||
|
||||
Review of different types of flexible joints for Stewart plaftorms (see Figure \ref{fig:detail_fem_joints_examples}).
|
||||
|
||||
@ -827,7 +748,7 @@ Typical values?
|
||||
\end{figure}
|
||||
|
||||
\section{Bending and Torsional Stiffness}
|
||||
\label{sec:org004c610}
|
||||
\label{sec:orgf11a334}
|
||||
\label{sec:joints_rot_stiffness}
|
||||
|
||||
Because of bending stiffness of the flexible joints, the forces applied by the struts are no longer aligned with the struts (additional forces applied by the ``spring force'' of the flexible joints).
|
||||
@ -908,7 +829,7 @@ Conclusion:
|
||||
\end{itemize}
|
||||
|
||||
\section{Axial Stiffness}
|
||||
\label{sec:org436b957}
|
||||
\label{sec:orgd08fa7c}
|
||||
\label{sec:joints_trans_stiffness}
|
||||
|
||||
\begin{itemize}
|
||||
@ -976,7 +897,7 @@ Conclusion:
|
||||
\end{itemize}
|
||||
|
||||
\section{Obtained design / Specifications}
|
||||
\label{sec:org1a780d9}
|
||||
\label{sec:org93383a9}
|
||||
|
||||
\begin{itemize}
|
||||
\item Summary of specifications (Table \ref{tab:detail_fem_joints_specs})
|
||||
@ -1036,7 +957,7 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
|
||||
\end{figure}
|
||||
|
||||
\section{Validation with the Nano-Hexapod}
|
||||
\label{sec:org6bcd4cf}
|
||||
\label{sec:org9751c8e}
|
||||
|
||||
To validate the designed flexible joint:
|
||||
\begin{itemize}
|
||||
@ -1110,7 +1031,7 @@ Talk about model order:
|
||||
\end{figure}
|
||||
|
||||
\chapter*{Conclusion}
|
||||
\label{sec:org14441b2}
|
||||
\label{sec:org57f9ca5}
|
||||
\label{sec:detail_fem_conclusion}
|
||||
|
||||
\printbibliography[heading=bibintoc,title={Bibliography}]
|
||||
|
||||
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Reference in New Issue
Block a user