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# Finite Element Model with Simscape
Author: Dehaeze Thomas

## Table of Contents

- 1. Amplified Piezoelectric Actuator - 3D elements
  - 1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
  - 1.2. Identification of the Dynamics
  - 1.3. Comparison with Ansys
  - 1.4. Force Sensor
  - 1.5. Distributed Actuator
  - 1.6. Distributed Actuator and Force Sensor
- 2. Integral Force Feedback with Amplified Piezo
  - 2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
  - 2.2. IFF Plant
  - 2.3. IFF controller
  - 2.4. Closed Loop System

## 1 Amplified Piezoelectric Actuator - 3D elements

The idea here is to:

- export a FEM of an amplified piezoelectric actuator from Ansys to Matlab
- import it into a Simscape model
- compare the obtained dynamics
- add 10kg mass on top of the actuator and identify the dynamics
- compare with results from Ansys where 10kg are directly added to the FEM

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

We first extract the stiffness and mass matrices.

    K = extractMatrix('piezo_amplified_3d_K.txt');
    M = extractMatrix('piezo_amplified_3d_M.txt');

Then, we extract the coordinates of the interface nodes.

    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_3d.txt');

Total number of Nodes | 168959
Number of interface Nodes | 13
Number of Modes | 30
Size of M and K matrices | 108

Figure 1: Interface Nodes for the Amplified Piezo Actuator

Table 1: Coordinates of the interface nodes
Node i | Node Number | x [m] | y [m] | z [m]
--- | --- | --- | --- | ---
1.0 | 168947.0 | 0.0 | 0.03 | 0.0
2.0 | 168949.0 | 0.0 | -0.03 | 0.0
3.0 | 168950.0 | -0.035 | 0.0 | 0.0
4.0 | 168951.0 | -0.028 | 0.0 | 0.0
5.0 | 168952.0 | -0.021 | 0.0 | 0.0
6.0 | 168953.0 | -0.014 | 0.0 | 0.0
7.0 | 168954.0 | -0.007 | 0.0 | 0.0
8.0 | 168955.0 | 0.0 | 0.0 | 0.0
9.0 | 168956.0 | 0.007 | 0.0 | 0.0
10.0 | 168957.0 | 0.014 | 0.0 | 0.0
11.0 | 168958.0 | 0.021 | 0.0 | 0.0
12.0 | 168959.0 | 0.035 | 0.0 | 0.0
13.0 | 168960.0 | 0.028 | 0.0 | 0.0

Table 2: First 10x10 elements of the Stiffness matrix
300000000.0 | -30000.0 | 8000.0 | -200.0 | -30.0 | -60000.0 | 20000000.0 | -4000.0 | 500.0 | 8
-30000.0 | 100000000.0 | 400.0 | 30.0 | 200.0 | -1 | 4000.0 | -8000000.0 | 800.0 | 7
8000.0 | 400.0 | 50000000.0 | -800000.0 | -300.0 | -40.0 | 300.0 | 100.0 | 5000000.0 | 40000.0
-200.0 | 30.0 | -800000.0 | 20000.0 | 5 | 1 | -10.0 | -2 | -40000.0 | -300.0
-30.0 | 200.0 | -300.0 | 5 | 40000.0 | 0.3 | -4 | -10.0 | 40.0 | 0.4
-60000.0 | -1 | -40.0 | 1 | 0.3 | 3000.0 | 7000.0 | 0.8 | -1 | 0.0003
20000000.0 | 4000.0 | 300.0 | -10.0 | -4 | 7000.0 | 300000000.0 | 20000.0 | 3000.0 | 80.0
-4000.0 | -8000000.0 | 100.0 | -2 | -10.0 | 0.8 | 20000.0 | 100000000.0 | -4000.0 | -100.0
500.0 | 800.0 | 5000000.0 | -40000.0 | 40.0 | -1 | 3000.0 | -4000.0 | 50000000.0 | 800000.0
8 | 7 | 40000.0 | -300.0 | 0.4 | 0.0003 | 80.0 | -100.0 | 800000.0 | 20000.0

Table 3: First 10x10 elements of the Mass matrix
0.03 | 2e-06 | -2e-07 | 1e-08 | 2e-08 | 0.0002 | -0.001 | 2e-07 | -8e-08 | -9e-10
2e-06 | 0.02 | -5e-07 | 7e-09 | 3e-08 | 2e-08 | -3e-07 | 0.0003 | -1e-08 | 1e-10
-2e-07 | -5e-07 | 0.02 | -9e-05 | 4e-09 | -1e-08 | 2e-07 | -2e-08 | -0.0006 | -5e-06
1e-08 | 7e-09 | -9e-05 | 1e-06 | 6e-11 | 4e-10 | -1e-09 | 3e-11 | 5e-06 | 3e-08
2e-08 | 3e-08 | 4e-09 | 6e-11 | 1e-06 | 2e-10 | -2e-09 | 2e-10 | -7e-09 | -4e-11
0.0002 | 2e-08 | -1e-08 | 4e-10 | 2e-10 | 2e-06 | -2e-06 | -1e-09 | -7e-10 | -9e-12
-0.001 | -3e-07 | 2e-07 | -1e-09 | -2e-09 | -2e-06 | 0.03 | -2e-06 | -1e-07 | -5e-09
2e-07 | 0.0003 | -2e-08 | 3e-11 | 2e-10 | -1e-09 | -2e-06 | 0.02 | -8e-07 | -1e-08
-8e-08 | -1e-08 | -0.0006 | 5e-06 | -7e-09 | -7e-10 | -1e-07 | -8e-07 | 0.02 | 9e-05
-9e-10 | 1e-10 | -5e-06 | 3e-08 | -4e-11 | -9e-12 | -5e-09 | -1e-08 | 9e-05 | 1e-06

Using `K`, `M` and `int_xyz`, we can use the `Reduced Order Flexible Solid` simscape block.

### 1.2 Identification of the Dynamics

The flexible element is imported using the `Reduced Order Flexible Solid` simscape block.

To model the actuator, an `Internal Force` block is added between the nodes 3 and 12. A `Relative Motion Sensor` block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.

One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.

We first set the mass to be zero.

    m = 0;

The dynamics is identified from the applied force to the measured relative displacement.

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;

    Gh = linearize(mdl, io);

Then, we add 10Kg of mass:

    m = 10;

And the dynamics is identified.

The two identified dynamics are compared in Figure 2.

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;

    Ghm = linearize(mdl, io);

Figure 2: Dynamics from \(F\) to \(d\) without a payload and with a 10kg payload

### 1.3 Comparison with Ansys

Let’s import the results from an Harmonic response analysis in Ansys.

    Gresp0 = readtable('FEA_HarmResponse_00kg.txt');
    Gresp10 = readtable('FEA_HarmResponse_10kg.txt');

The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure 3.

Figure 3: Comparison of the obtained dynamics using Simscape with the harmonic response analysis using Ansys

### 1.4 Force Sensor

The dynamics is identified from internal forces applied between nodes 3 and 11 to the relative displacement of nodes 11 and 13.

The obtained dynamics is shown in Figure 4.

    m = 0;

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;

    Gf = linearize(mdl, io);

    m = 10;

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;

    Gfm = linearize(mdl, io);

Figure 4: Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)

### 1.5 Distributed Actuator

    m = 0;

The dynamics is identified from the applied force to the measured relative displacement.

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d_distri';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;

    Gd = linearize(mdl, io);

Then, we add 10Kg of mass:

    m = 10;

And the dynamics is identified.

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d_distri';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;

    Gdm = linearize(mdl, io);

### 1.6 Distributed Actuator and Force Sensor

    m = 0;

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d_distri_act_sens';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;

    Gfd = linearize(mdl, io);

    m = 10;

    %% Name of the Simulink File
    mdl = 'piezo_amplified_3d_distri_act_sens';

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;

    Gfdm = linearize(mdl, io);

## 2 Integral Force Feedback with Amplified Piezo

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

We first extract the stiffness and mass matrices.

    K = extractMatrix('piezo_amplified_IFF_K.txt');
    M = extractMatrix('piezo_amplified_IFF_M.txt');

Then, we extract the coordinates of the interface nodes.

    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_IFF.txt');

### 2.2 IFF Plant

The transfer function from the force actuator to the force sensor is identified and shown in Figure 5.

    Kiff = tf(0);

    m = 0;

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Kiff'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/G'],    1, 'openoutput'); io_i = io_i + 1;

    Gf = linearize(mdl, io);

    m = 10;

    Gfm = linearize(mdl, io);

Figure 5: IFF Plant

### 2.3 IFF controller

The controller is defined and the loop gain is shown in Figure 6.

    Kiff = -1e12/s;

Figure 6: IFF Loop Gain

### 2.4 Closed Loop System

    m = 10;

    Kiff = -1e12/s;

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

    %% Input/Output definition
    clear io; io_i = 1;
    io(io_i) = linio([mdl, '/Dw'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
    io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1;
    io(io_i) = linio([mdl, '/G'],  1, 'output'); io_i = io_i + 1;

    Giff = linearize(mdl, io);
    Giff.InputName  = {'w', 'f', 'F'};
    Giff.OutputName  = {'x1', 'Fs'};

    Kiff = tf(0);

    G = linearize(mdl, io);
    G.InputName  = {'w', 'f', 'F'};
    G.OutputName  = {'x1', 'Fs'};

Figure 7: OL and CL transfer functions

Figure 8: Results obtained in souleille18_concep_activ_mount_space_applic

Author: Dehaeze Thomas

Created: 2020-06-15 lun. 09:13