start writing first section

nass-fem.org

@@ -1102,7 +1102,15 @@ CLOSED: [2025-02-25 Tue 00:17]
 - [X] Should be able to use =ga= and =gs=
 - [X] Look at what is done in C6 section
 
-** TODO [#A] Little review of flexible joints for spherical and universal joints
+** TODO [#A] Check all figures
+
+- [ ] Legend
+- [ ] Units
+- [ ] Notations
+- [ ] ...
+
+** DONE [#A] Little review of flexible joints for spherical and universal joints
+CLOSED: [2025-02-26 Wed 10:13]
 
 For Stewart platform:
 - ID16a [[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]]
@@ -1131,16 +1139,17 @@ For Stewart platform:
 <<sec:detail_fem_super_element>>
 ** Introduction                                                      :ignore:
 
-Goal:
+Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
-- include parts from which dynamical properties are estimated from a FEM
+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:
+The investigation of this hybrid modeling approach is structured in three sections.
-- Quick explanation of the theory
+First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section ref:ssec:detail_fem_super_element_theory).
-- Explain the implementation with FEA software (Ansys) and Simscape
+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).
-- Experimental validation with an amplified piezoelectric actuator
+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).
 
-[[cite:&rankers98_machin]]
+The work presented in this section has also been published in [[cite:&brumund21_multib_simul_reduc_order_flexib_bodies_fea]].
-[[cite:&hatch00_vibrat_matlab_ansys]]
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1173,43 +1182,40 @@ open(mdl); % Open Simscape Model
 <<m-init-other>>
 #+end_src
 
-** FEA Modal Reduction
+** Procedure
 <<ssec:detail_fem_super_element_theory>>
 
-- 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
+In this modeling approach, some components within the multi-body framework are represented as /reduced-order flexible bodies/, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models.
-- matrices were created from FEA models via modal reduction techniques, more specifically the component mode synthesis (CMS).
+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.
-- this makes this design approach a combined multibody-FEA technique.
 
+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.
 
-- FEM: high number of DoF
+The procedure for implementing this reduction involves several distinct stages.
-- goal: reduce number of DoF, allow to integrate in multi-body simulation
+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:
+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.
-- model the part in FE software as usually by defining material properties, etc.
+This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF.
-- define frames for which we want to the multi-body model will then be able to interface with, and can be used to:
+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.
-  - connect other parts
+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.
-  - apply forces and torques
-  - measure motion between frames
-- 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
-- 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
-- then, it outputs $m \times m$ reduced mass and stiffness matrices
-- in the multi-body model, the two reduced matrices can be used to model the part
 
-** Validation of the Method
+\begin{equation}\label{eq:detail_fem_model_order}
-<<ssec:detail_fem_super_element_validation>>
+m = 6 \times n + p
+\end{equation}
+
+** Example with an Amplified Piezoelectric Actuator
+<<ssec:detail_fem_super_element_example>>
 **** Introduction                                                  :ignore:
 
-Validation with Amplified Piezoelectric Actuator, because:
+The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons.
-- is a good candidate for the nano-hexapod (as will be explained in Section ref:sec:detail_fem_actuator)
+Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section ref:sec:detail_fem_actuator.
-- had one in the lab for experimental testing (APA95ML, Figure ref:fig:detail_fem_apa95ml_picture)
+Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing.
-  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.
-- permits to model a mechanical structure (similar to a flexible joint), piezoelectric actuator and piezoelectric sensor
 
-Quick explanation of APA:
+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]].
-- [[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.
 
 #+attr_latex: :options [b]{0.48\linewidth}
 #+begin_minipage
@@ -1222,30 +1228,20 @@ Quick explanation of APA:
 #+attr_latex: :options [b]{0.48\linewidth}
 #+begin_minipage
 #+latex: \centering
-#+attr_latex: :environment tabularx :width 0.8\linewidth :placement [b] :align Xcc
+#+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align Xc
 #+attr_latex: :booktabs t :float nil :center nil
-| Parameter      | Unit      | Value |
+| *Parameter*    | *Value*       |
-|----------------+-----------+-------|
+|----------------+---------------|
-| Nominal Stroke | $\mu m$   |   100 |
+| Nominal Stroke | $100\,\mu m$  |
-| Blocked force  | $N$       |  1600 |
+| Blocked force  | $1600\,N$     |
-| Stiffness      | $N/\mu m$ |    16 |
+| Stiffness      | $16\,N/\mu m$ |
 #+latex: \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
 #+end_minipage
 
 **** Finite Element Model
 
-- explain how the FEM is done:
+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.
-  - material properties (Table ref:tab:detail_fem_material_properties)
+The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was then generated.
-  - mesh (Figure ref:fig:detail_fem_apa95ml_mesh)
-- explain piezoelectric materials:
-  - sensors
-  - actuators
-- choice of frames (Figure ref:fig:detail_fem_apa95ml_frames)
-  - 2 for each piezoelectric stack to measure strain and apply forces
-  - 1 at the top, 1 at the bottom to connect to other elements
-- choose number of DoF => size of model
-  7 frames + 6 modes => order 48
-- perform the reduction: show the output reduced matrices
 
 #+name: tab:detail_fem_material_properties
 #+caption: Material properties used for FEA modal reduction model. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density
@@ -1256,6 +1252,12 @@ Quick explanation of APA:
 | Stainless Steel              | $190\,GPa$  | $0.31$ | $7800\,\text{kg}/m^3$ |
 | Piezoelectric Ceramics (PZT) | $49.5\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ |
 
+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.
+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.
+
+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.
+
 #+name: fig:detail_fem_apa95ml_model
 #+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}).
 #+attr_latex: :options [htbp]
@@ -1276,6 +1278,8 @@ Quick explanation of APA:
 
 **** Super Element in the Multi-Body Model
 
+
+
 Model:
 - Connect frame $\{4\}$ to world frame and frame $\{6\}$ to a 5.5kg mass, vertically guided
 - 2 actuator stacks, 1 sensor stack:
@@ -1375,7 +1379,10 @@ ka  = cE*A/L;    % Stiffness of the two stacks [N/m]
 ga = d33*n*ka;   % Actuator Constant [N/V]
 #+end_src
 
-**** Experimental Validation
+** Experimental Validation
+<<ssec:detail_fem_super_element_validation>>
+**** Introduction                                                  :ignore:
+**** Test Bench
 
 goal: validation of the procedure.
 
@@ -1403,6 +1410,8 @@ goal: validation of the procedure.
 #+end_subfigure
 #+end_figure
 
+**** 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 $V_s$
@@ -1715,8 +1724,6 @@ exportFig('figs/detail_fem_apa95ml_damped_plants.pdf', 'width', 'half', 'height'
 - 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
 
-- [ ] [[file:~/Cloud/research/papers/published/brumund21_multib_simul_reduc_order_flexib_bodies_fea/paper/brumund21_multib_simul_reduc_order_flexib_bodies_fea.pdf][published paper]]
-
 * Actuator
 <<sec:detail_fem_actuator>>
 ** Introduction                                                      :ignore:
@@ -3526,7 +3533,6 @@ colors = colororder;
 freqs = logspace(1,3,500); % Frequency vector [Hz]
 #+END_SRC
 
-
 * Matlab Functions                                                  :noexport:
 *** =initializeSimplifiedNanoHexapod=: Nano Hexapod
 

nass-fem.tex

@@ -1,4 +1,4 @@
-% Created 2025-02-25 Tue 19:07
+% Created 2025-02-26 Wed 09:37
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -43,6 +43,7 @@ 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:detail_fem_super_element}
 Goal:
 \begin{itemize}
@@ -59,6 +60,7 @@ Outline:
 \cite{rankers98_machin}
 \cite{hatch00_vibrat_matlab_ansys}
 \section{FEA Modal Reduction}
+\label{sec:org4844a44}
 \label{ssec:detail_fem_super_element_theory}
 
 \begin{itemize}
@@ -82,7 +84,7 @@ Procedure:
 \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) for the reduction of the high number of FEA degrees of freedom (DoF) to a smaller number of retained degrees of freedom
+\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
 \(m = 6 \times n + p\)
@@ -92,6 +94,7 @@ typically from hundred thousands to less than 100 DoF
 \end{itemize}
 
 \section{Validation of the Method}
+\label{sec:orgcb6472b}
 \label{ssec:detail_fem_super_element_validation}
 Validation with Amplified Piezoelectric Actuator, because:
 \begin{itemize}
@@ -127,6 +130,7 @@ Stiffness & \(N/\mu m\) & 16\\
 \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
 \end{minipage}
 \paragraph{Finite Element Model}
+\label{sec:orgf976215}
 
 \begin{itemize}
 \item explain how the FEM is done:
@@ -179,6 +183,7 @@ Piezoelectric Ceramics (PZT) & \(49.5\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\
 \end{figure}
 
 \paragraph{Super Element in the Multi-Body Model}
+\label{sec:orga1214e3}
 
 Model:
 \begin{itemize}
@@ -219,6 +224,7 @@ In order to correctly model the piezoelectric actuator with Simscape, the values
 \end{itemize}
 
 \paragraph{Sensor and Actuator ``constants''}
+\label{sec:orgb6b6d3f}
 
 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}).
 
@@ -274,6 +280,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
 \end{table}
 
 \paragraph{Experimental Validation}
+\label{sec:orgfd4d8f6}
 
 goal: validation of the procedure.
 
@@ -336,6 +343,7 @@ goal: validation of the procedure.
 \end{figure}
 
 \paragraph{Integral Force Feedback with APA}
+\label{sec:orgbd44486}
 
 goal:
 \begin{itemize}
@@ -373,6 +381,7 @@ In the frequency band of interest, this controller should mostly act as a pure i
 \end{figure}
 
 \section*{Conclusion}
+\label{sec:orga81cb67}
 \begin{itemize}
 \item Validation of the method
 \item Very useful to optimize different parts
@@ -384,6 +393,7 @@ In the frequency band of interest, this controller should mostly act as a pure i
 \end{itemize}
 
 \chapter{Actuator}
+\label{sec:orgece4287}
 \label{sec:detail_fem_actuator}
 Goals:
 \begin{itemize}
@@ -394,6 +404,7 @@ 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{ssec:detail_fem_actuator_specifications}
 
 From previous analysis:
@@ -504,6 +515,7 @@ Height  \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
 \end{table}
 
 \section{APA300ML - Reduced Order Flexible Body}
+\label{sec:orgcc3207d}
 \label{ssec:detail_fem_actuator_apa300ml}
 
 To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa300ml_picture}):
@@ -540,15 +552,18 @@ As the stacks are the same between the APA300ML and the APA95ML, the values esti
 \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}).
@@ -561,6 +576,7 @@ This is estimated at 709Hz from the model (Figure \ref{fig:detail_fem_apa300ml_c
 
 
 \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.
@@ -575,6 +591,7 @@ With an amplification factor equal to \(5\), the vertical stroke is estimated at
 This analysis provides some confidence on the model accuracy.
 
 \section{Simpler 2DoF Model of the APA300ML}
+\label{sec:org3340d21}
 \label{sec:apa_model}
 \begin{itemize}
 \item \emph{super-element} order is quite large, and therefore not practical for simulations
@@ -591,6 +608,7 @@ 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}
 
 The model is adapted from \cite{souleille18_concep_activ_mount_space_applic}.
 
@@ -620,6 +638,7 @@ The main advantage is that this model is very simple, only adds 4 states
 \end{figure}
 
 \paragraph{Parameter Tuning}
+\label{sec:org6d0757e}
 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}
@@ -675,6 +694,7 @@ 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}
 
 \begin{itemize}
 \item Mechanical equations and electrical equations are coupled
@@ -697,6 +717,7 @@ This will be discussed in chapter ``instrumentation''
 \end{figure}
 
 \section{Validation with the Nano-Hexapod}
+\label{sec:org461915d}
 NASS model + FEM model (or just 2DoF) of APA300ML => validation (based on what?)
 
 \begin{itemize}
@@ -734,6 +755,7 @@ here matrices have a size of 36
 
 
 \chapter{Flexible Joint}
+\label{sec:orgfd42b09}
 \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}
@@ -762,6 +784,7 @@ 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}
 
 Review of different types of flexible joints for Stewart plaftorms (see Figure \ref{fig:detail_fem_joints_examples}).
 
@@ -804,6 +827,7 @@ Typical values?
 \end{figure}
 
 \section{Bending and Torsional Stiffness}
+\label{sec:org004c610}
 \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).
@@ -884,6 +908,7 @@ Conclusion:
 \end{itemize}
 
 \section{Axial Stiffness}
+\label{sec:org436b957}
 \label{sec:joints_trans_stiffness}
 
 \begin{itemize}
@@ -951,6 +976,7 @@ Conclusion:
 \end{itemize}
 
 \section{Obtained design / Specifications}
+\label{sec:org1a780d9}
 
 \begin{itemize}
 \item Summary of specifications (Table \ref{tab:detail_fem_joints_specs})
@@ -1010,6 +1036,7 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
 \end{figure}
 
 \section{Validation with the Nano-Hexapod}
+\label{sec:org6bcd4cf}
 
 To validate the designed flexible joint:
 \begin{itemize}
@@ -1083,6 +1110,7 @@ Talk about model order:
 \end{figure}
 
 \chapter*{Conclusion}
+\label{sec:org14441b2}
 \label{sec:detail_fem_conclusion}
 
 \printbibliography[heading=bibintoc,title={Bibliography}]