Finite Element Model with Simscape

1. Amplified Piezoelectric Actuator - 3D elements

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]
1.0	168947.0	0.0	0.03
2.0	168949.0	0.0	-0.03
3.0	168950.0	-0.035	0.0
4.0	168951.0	-0.028	0.0
5.0	168952.0	-0.021	0.0
6.0	168953.0	-0.014	0.0
7.0	168954.0	-0.007	0.0
8.0	168955.0	0.0	0.0
9.0	168956.0	0.007	0.0
10.0	168957.0	0.014	0.0
11.0	168958.0	0.021	0.0
12.0	168959.0	0.035	0.0
13.0	168960.0	0.028	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 first and second nodes. To model the sensors, a Relative Motion Sensor block is added between the second and the third nodes.

Two masses are fixed at the ends of the piezo-electric stack actuator.

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');

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

1.4 Force Sensor

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 \(m = 10kg\)