diff --git a/index.html b/index.html index 80b166b..0f8c37d 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + NASS - Finite Element Models with Simscape @@ -30,46 +30,47 @@

Table of Contents

@@ -84,22 +85,22 @@ In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are d It is divided in the following sections:

-
-

1 APA300ML

+
+

1 APA300ML

- +

In this section, the Amplified Piezoelectric Actuator APA300ML (doc) is modeled using a Finite Element Software. @@ -107,19 +108,19 @@ Then a super element is exported and imported in Simscape where its dynam

-A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure 1. +A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure 1. The remote point used are also shown in this figure.

-
+

apa300ml_ansys.jpg

Figure 1: Ansys FEM of the APA300ML

-
-

1.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

+
+

1.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

We first extract the stiffness and mass matrices. @@ -573,14 +574,14 @@ Using K, M and int_xyz, we can now use th

-
-

1.2 Piezoelectric parameters

+
+

1.2 Piezoelectric parameters

In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:

-
d33 = 300e-12; % Strain constant [m/V]
+
d33 = 600e-12; % Strain constant [m/V]
 n   = 80;      % Number of layers per stack
 eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
 sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
@@ -589,11 +590,24 @@ C   = 5e-6;    % Stack c
+

+PZT-4 +

+
+
d33 = 300e-12; % Strain constant [m/V]
+n   = 80;      % Number of layers per stack
+eT  = 5.3e-9;  % Permittivity under constant stress [F/m]
+sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
+ka  = 235e6;   % Stack stiffness [N/m]
+C   = 5e-6;    % Stack capactiance [F]
+
+
+

The ratio of the developed force to applied voltage is:

\begin{equation} -\label{org26cf049} +\label{orgbc04a1b} F_a = g_a V_a, \quad g_a = d_{33} n k_a \end{equation}

@@ -624,7 +638,7 @@ If we take the numerical values, we obtain: From (Fleming and Leang 2014) (page 123), the relation between relative displacement of the sensor stack and generated voltage is:

\begin{equation} -\label{orgd71c6e4} +\label{orgb1b83fa} V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \end{equation}

@@ -653,8 +667,8 @@ If we take the numerical values, we obtain:

-
-

1.3 Simscape Model

+
+

1.3 Simscape Model

The flexible element is imported using the Reduced Order Flexible Solid simscape block. @@ -669,7 +683,7 @@ Let’s say we use two stacks as a force sensor and one stack as an actuator

-The interface nodes are shown in Figure 1. +The interface nodes are shown in Figure 1.

@@ -678,12 +692,12 @@ One mass is fixed at one end of the piezo-electric stack actuator (remove point

-
-

1.4 Identification of the APA Characteristics

+
+

1.4 Identification of the APA Characteristics

-
-

1.4.1 Stiffness

+
+

1.4.1 Stiffness

The transfer function from vertical external force to the relative vertical displacement is identified. @@ -708,16 +722,16 @@ The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).

-
-

1.4.2 Resonance Frequency

+
+

1.4.2 Resonance Frequency

The resonance frequency is specified to be between 650Hz and 840Hz. -This is also the case for the FEM model (Figure 2). +This is also the case for the FEM model (Figure 2).

-
+

apa300ml_resonance.png

Figure 2: First resonance is around 800Hz

@@ -725,8 +739,8 @@ This is also the case for the FEM model (Figure 2).
-
-

1.4.3 Amplification factor

+
+

1.4.3 Amplification factor

The amplification factor is the ratio of the vertical displacement to the stack displacement. @@ -759,8 +773,8 @@ This is actually correct and approximately corresponds to the ratio of the piezo

-
-

1.4.4 Stroke

+
+

1.4.4 Stroke

Estimation of the actuator stroke: @@ -789,10 +803,19 @@ This is exactly the specified stroke in the data-sheet.

+ +
+

1.4.5 Stroke BIS

+
+
    +
  • [ ] Identified the stroke form the transfer function from V to z
    • +
+
+
-
-

1.5 Identification of the Dynamics from actuator to replace displacement

+
+

1.5 Identification of the Dynamics from actuator to replace displacement

We first set the mass to be approximately zero. @@ -805,17 +828,17 @@ The same dynamics is identified for a payload mass of 10Kg.

-
+

apa300ml_plant_dynamics.png

Figure 3: Transfer function from forces applied by the stack to the axial displacement of the APA

-The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure 4. +The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure 4.

-
+

apa300ml_dvf_root_locus.png

Figure 4: Root Locus for Direct Velocity Feedback

@@ -823,28 +846,28 @@ The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is
-
-

1.6 Identification of the Dynamics from actuator to force sensor

+
+

1.6 Identification of the Dynamics from actuator to force sensor

Let’s use 2 stacks as a force sensor and 1 stack as force actuator.

-The transfer function from actuator voltage to sensor voltage is identified and shown in Figure 5. +The transfer function from actuator voltage to sensor voltage is identified and shown in Figure 5.

-
+

apa300ml_iff_plant.png

Figure 5: Transfer function from actuator to force sensor

-For root locus corresponding to IFF is shown in Figure 6. +For root locus corresponding to IFF is shown in Figure 6.

-
+

apa300ml_iff_root_locus.png

Figure 6: Root Locus for IFF

@@ -852,8 +875,8 @@ For root locus corresponding to IFF is shown in Figure 6
-
-

1.7 Identification for a simpler model

+
+

1.7 Identification for a simpler model

The goal in this section is to identify the parameters of a simple APA model from the FEM. @@ -865,12 +888,12 @@ The presented model is based on (Souleille et al.

-The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 7). +The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 7). The parameters are shown in the table below.

-
+

souleille18_model_piezo.png

Figure 7: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator

@@ -1019,11 +1042,11 @@ And the DC gain is adjusted for the force sensor:

-The dynamics of the FEM model and the simpler model are compared in Figure 8. +The dynamics of the FEM model and the simpler model are compared in Figure 8.

-
+

apa300ml_comp_simpler_model.png

Figure 8: Comparison of the Dynamics between the FEM model and the simplified one

@@ -1034,10 +1057,10 @@ The simplified model has also been implemented in Simscape.

-The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure 9. +The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure 9.

-
+

apa300ml_comp_simpler_simscape.png

Figure 9: Comparison of the Dynamics between the FEM model and the simplified simscape model

@@ -1045,8 +1068,8 @@ The dynamics of the Simscape simplified model is identified and compared with th
-
-

1.8 Integral Force Feedback

+
+

1.8 Integral Force Feedback

In this section, Integral Force Feedback control architecture is applied on the APA300ML. @@ -1062,18 +1085,18 @@ The payload mass is set to 10kg.

-The obtained dynamics is shown in Figure 10. +The obtained dynamics is shown in Figure 10.

-
+

piezo_amplified_iff_plant.png

Figure 10: IFF Plant

-The controller is defined below and the loop gain is shown in Figure 11. +The controller is defined below and the loop gain is shown in Figure 11. 

Kiff = -1e3/s;
@@ -1081,29 +1104,29 @@ The controller is defined below and the loop gain is shown in Figure 
+

 
piezo_amplified_iff_loop_gain.png
 

 
Figure 11: IFF Loop Gain

 

 
 

-Now the closed-loop system is identified again and compare with the open loop system in Figure 12.
+Now the closed-loop system is identified again and compare with the open loop system in Figure 12.
 

 
 

-It is the expected behavior as shown in the Figure 13 (from (Souleille et al. 2018)).
+It is the expected behavior as shown in the Figure 13 (from (Souleille et al. 2018)).
 

 
 
-

+

 
piezo_amplified_iff_comp.png
 

 
Figure 12: OL and CL transfer functions

 

 
 
-

+

 
souleille18_results.png
 

 
Figure 13: Results obtained in souleille18_concep_activ_mount_space_applic

@@ -1113,14 +1136,14 @@ It is the expected behavior as shown in the Figure 13
 

 
 
-

-
2 First Flexible Joint Geometry

+

+
2 First Flexible Joint Geometry

 

 

-
+
 

 

-The studied flexor is shown in Figure 14.
+The studied flexor is shown in Figure 14.
 

 
 

@@ -1133,14 +1156,14 @@ A simplified model of the flexor is then developped.
 

 
 
-

+

 
flexor_id16_screenshot.png
 

 
Figure 14: Flexor studied

 

 

-

-
2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

+

+
2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

 

 

 We first extract the stiffness and mass matrices.
@@ -1552,8 +1575,8 @@ Using K, M and int_xyz, we can use the 
 

 
-

-
2.2 Identification of the parameters using Simscape and looking at the Stiffness Matrix

+

+
2.2 Identification of the parameters using Simscape and looking at the Stiffness Matrix

 

 

 The flexor is now imported into Simscape and its parameters are estimated using an identification.
@@ -1610,15 +1633,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
 

 

 
-

-
2.3 Simpler Model

+

+
2.3 Simpler Model

 

 

-Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 15.
+Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 15.
 

 
 
-

+

 
flexible_joint_simscape.png
 

 
Figure 15: Bushing Joint used to model the flexible joint

@@ -1643,7 +1666,7 @@ The two obtained dynamics are compared in Figure
 

 
 
-

+

 
flexor_ID16_compare_bushing_joint.png
 

 
Figure 16: Comparison of the Joint compliance between the FEM model and the simpler model

@@ -1652,29 +1675,29 @@ The two obtained dynamics are compared in Figure
 

 

 
-

-
3 Optimized Flexible Joint

+

+
3 Optimized Flexible Joint

 

 

-
+
 

 

 The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.
 

 
 

-The obtained geometry is shown in Figure 17.
+The obtained geometry is shown in Figure 17.
 

 
 
-

+

 
flexor_025_MDoF.jpg
 

 
Figure 17: Flexor studied

 

 

-

-
3.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

+

+
3.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

 

 

 We first extract the stiffness and mass matrices.
@@ -2088,8 +2111,8 @@ Using K, M and int_xyz, we can use the 
 

 
-

-
3.2 Identification of the parameters using Simscape

+

+
3.2 Identification of the parameters using Simscape

 

 

 The flexor is now imported into Simscape and its parameters are estimated using an identification.
@@ -2146,15 +2169,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
 

 

 
-

-
3.3 Simpler Model

+

+
3.3 Simpler Model

 

 

-Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 15.
+Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 15.
 

 
 
-

+

 
flexible_joint_simscape.png
 

 
Figure 18: Bushing Joint used to model the flexible joint

@@ -2179,7 +2202,7 @@ The two obtained dynamics are compared in Figure
 

 
 
-

+

 
flexor_ID16_compare_bushing_joint.png
 

 
Figure 19: Comparison of the Joint compliance between the FEM model and the simpler model

@@ -2187,8 +2210,8 @@ The two obtained dynamics are compared in Figure
 

 

 
-

-
3.4 Comparison with a stiffer Flexible Joint

+

+
3.4 Comparison with a stiffer Flexible Joint

 

 

 The stiffness matrix with the flexible joint with a “hinge” size of 0.50mm is loaded.
@@ -2255,38 +2278,38 @@ Its parameters are compared with the Flexible Joint with a size of 0.25mm in the
 

 

 
-

-
4 Complete Strut with Encoder

+

+
4 Complete Strut with Encoder

 

 

-
+
 

 

-

-
4.1 Introduction

+

+
4.1 Introduction

 

 

 Now, the full nano-hexapod strut is modelled using Ansys.
 

 
 

-The 3D as well as the interface nodes are shown in Figure 20.
+The 3D as well as the interface nodes are shown in Figure 20.
 

 
 
-

-
strut_encoder_nodes.jpg
+

+
strut_fem_nodes.jpg
 

 
Figure 20: Interface points

 

 
 

-A side view is shown in Figure 21.
+A side view is shown in Figure 21.
 

 
 
-

-
strut_encoder_nodes_side.jpg
+

+
strut_fem_nodes_side.jpg
 

 
Figure 21: Interface points - Side view

 

@@ -2297,8 +2320,8 @@ The flexible joints used have a 0.25mm width size.
 

 

 
-

-
4.2 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

+

+
4.2 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

 

 

 We first extract the stiffness and mass matrices.
@@ -2760,20 +2783,20 @@ Using K, M and int_xyz, we can use the 
 

 
-

-
4.3 Piezoelectric parameters

+

+
4.3 Piezoelectric parameters

 

 

 Parameters for the APA300ML:
 

 
 

-
d33 = 3e-10; % Strain constant [m/V]
-n = 80; % Number of layers per stack
-eT = 1.6e-8; % Permittivity under constant stress [F/m]
-sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
-ka = 235e6; % Stack stiffness [N/m]
-C = 5e-6; % Stack capactiance [F]
+
d33 = 300e-12; % Strain constant [m/V]
+n   = 80;      % Number of layers per stack
+eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
+sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
+ka  = 235e6;   % Stack stiffness [N/m]
+C   = 5e-6;    % Stack capactiance [F]
 

 

 
@@ -2785,8 +2808,8 @@ ns = 1; % Number of stacks used as force sensor
 

 

 
-

-
4.4 Identification of the Dynamics

+

+
4.4 Identification of the Dynamics

 

 

 The dynamics is identified from the applied force to the measured relative displacement.
@@ -2798,7 +2821,7 @@ The same dynamics is identified for a payload mass of 10Kg.
 

 
 
-

+

 
dynamics_encoder_full_strut.png
 

 
Figure 22: Dynamics from the force actuator to the measured motion by the encoder

@@ -2819,7 +2842,7 @@ The same dynamics is identified for a payload mass of 10Kg.
 

 

 
Author: Dehaeze Thomas

-
Created: 2020-11-13 ven. 08:56

+
Created: 2021-01-04 lun. 13:57

 

 
 
diff --git a/index.org b/index.org
index 55c156b..a31fafe 100644
--- a/index.org
+++ b/index.org
@@ -168,7 +168,7 @@ Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid=
 ** Piezoelectric parameters
 In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
 #+begin_src matlab
-  d33 = 300e-12; % Strain constant [m/V]
+  d33 = 600e-12; % Strain constant [m/V]
   n   = 80;      % Number of layers per stack
   eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
   sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
@@ -176,6 +176,16 @@ In order to make the conversion from applied voltage to generated force or from
   C   = 5e-6;    % Stack capactiance [F]
 #+end_src
 
+PZT-4
+#+begin_src matlab
+  d33 = 300e-12; % Strain constant [m/V]
+  n   = 80;      % Number of layers per stack
+  eT  = 5.3e-9;  % Permittivity under constant stress [F/m]
+  sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
+  ka  = 235e6;   % Stack stiffness [N/m]
+  C   = 5e-6;    % Stack capactiance [F]
+#+end_src
+
 The ratio of the developed force to applied voltage is:
 #+name: eq:piezo_voltage_to_force
 \begin{equation}
@@ -231,7 +241,7 @@ One mass is fixed at one end of the piezo-electric stack actuator (remove point
 ** Identification of the APA Characteristics
 *** Stiffness
 #+begin_src matlab :exports none
-  m = 0.001;
+  m = 0.0001;
 #+end_src
 
 The transfer function from vertical external force to the relative vertical displacement is identified.
@@ -334,6 +344,23 @@ with:
 
 This is exactly the specified stroke in the data-sheet.
 
+*** TODO Stroke BIS
+- [ ] Identified the stroke form the transfer function from V to z
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/V'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
+
+  G = linearize(mdl, io);
+
+  1e6*170*abs(dcgain(G))
+#+end_src
+
 ** Identification of the Dynamics from actuator to replace displacement
 We first set the mass to be approximately zero.
 #+begin_src matlab :exports none
@@ -1495,13 +1522,13 @@ The 3D as well as the interface nodes are shown in Figure [[fig:strut_encoder_po
 
 #+name: fig:strut_encoder_points3
 #+caption: Interface points
-[[file:figs/strut_encoder_nodes.jpg]]
+[[file:figs/strut_fem_nodes.jpg]]
 
 A side view is shown in Figure [[fig:strut_encoder_nodes_side]].
 
 #+name: fig:strut_encoder_nodes_side
 #+caption: Interface points - Side view
-[[file:figs/strut_encoder_nodes_side.jpg]]
+[[file:figs/strut_fem_nodes_side.jpg]]
 
 The flexible joints used have a 0.25mm width size.
 
@@ -1608,12 +1635,12 @@ Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= sims
 Parameters for the APA300ML:
 
 #+begin_src matlab
-  d33 = 3e-10; % Strain constant [m/V]
-  n = 80; % Number of layers per stack
-  eT = 1.6e-8; % Permittivity under constant stress [F/m]
-  sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
-  ka = 235e6; % Stack stiffness [N/m]
-  C = 5e-6; % Stack capactiance [F]
+  d33 = 300e-12; % Strain constant [m/V]
+  n   = 80;      % Number of layers per stack
+  eT  = 1.6e-8;  % Permittivity under constant stress [F/m]
+  sD  = 1e-11;   % Compliance under constant electric displacement [m2/N]
+  ka  = 235e6;   % Stack stiffness [N/m]
+  C   = 5e-6;    % Stack capactiance [F]
 #+end_src
 
 #+begin_src matlab