Finite Element Model with Simscape

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');

save('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

load('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

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

Parameters for the APA95ML:

d33 = 3e-10; % Strain constant [m/V]
n = 80; % Number of layers per stack
eT = 1.6e-7; % 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]

na = 2; % Number of stacks used as actuator
ns = 1; % Number of stacks used as force sensor

The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V]. We denote this constant by \(g_a\) and: \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]

d33*(na*n)*(ka/(na + ns)) % [N/V]

3.76

From (Fleming and Leang 2014) (page 123), the relation between relative displacement and generated voltage is: \[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \] where:

• \(V_s\): measured voltage [V]
• \(d_{33}\): strain constant [m/V]
• \(\epsilon^T\): permittivity under constant stress [F/m]
• \(s^D\): elastic compliance under constant electric displacement [m^2/N]
• \(n\): number of layers
• \(\Delta h\): relative displacement [m]

1e-6*d33/(eT*sD*ns*n) % [V/um]

1.1719

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 = 0.01;

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 = 5;

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);

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.

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);

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);

m = 5;

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, '/V'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;

G = -linearize(mdl, io);

save('../test-bench-apa/mat/fem_model_5kg.mat', 'G')

The goal in this section is to identify the parameters of a simple APA model from the FEM. This can be useful is a lower order model is to be used for simulations.

The presented model is based on (Souleille et al. 2018).

The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 5). The parameters are shown in the table below.

The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.

\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] \[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]

If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with: \[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \] \[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]

From the identified dynamics, compute \(\alpha\) and \(\beta\)

alpha = abs(dcgain(G('y', 'Fa')));
beta  = abs(dcgain(G('y', 'Fd')));

\(k_a\) is estimated using the following formula:

ka = 0.9/abs(dcgain(G('y', 'Fa')));

The factor can be adjusted to better match the curves.

Then \(k_e\) and \(k_1\) are computed.

ke = ka/(beta/alpha - 1);
k1 = 1/beta - ke*ka/(ke + ka);

The damping in the system is adjusted to match the FEM model if necessary.

c1 = 1e2;

Analytical model of the simpler system:

Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
     [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
      -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];

Ga.InputName = {'Fd', 'w', 'Fa'};
Ga.OutputName = {'y', 'Fs'};

Adjust the DC gain for the force sensor:

F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));

We save the parameters of the simplified model for the APA95ML:

save('./mat/APA95ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');

We first extract the stiffness and mass matrices.

K = readmatrix('mat_K.CSV');
M = readmatrix('mat_M.CSV');

Then, we extract the coordinates of the interface nodes.

[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');

save('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

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

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]

na = 2; % Number of stacks used as actuator
ns = 1; % Number of stacks used as force sensor

The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V]. We denote this constant by \(g_a\) and: \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]

d33*(na*n)*(ka/(na + ns)) % [N/V]

3.76

From (Fleming and Leang 2014) (page 123), the relation between relative displacement and generated voltage is: \[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \] where:

• \(V_s\): measured voltage [V]
• \(d_{33}\): strain constant [m/V]
• \(\epsilon^T\): permittivity under constant stress [F/m]
• \(s^D\): elastic compliance under constant electric displacement [m^2/N]
• \(n\): number of layers
• \(\Delta h\): relative displacement [m]

1e-6*d33/(eT*sD*ns*n) % [V/um]

11.719

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. The dynamics is identified from the applied force to the measured relative displacement. The same dynamics is identified for a payload mass of 10Kg.

m = 10;

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

The transfer function from actuator to sensors is identified and shown in Figure 10.

For root locus corresponding to IFF is shown in Figure 11.

Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure 12.

The root locus for DVF is shown in Figure 13.

The goal in this section is to identify the parameters of a simple APA model from the FEM. This can be useful is a lower order model is to be used for simulations.

The presented model is based on (Souleille et al. 2018).

The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 5). The parameters are shown in the table below.

The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.

\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] \[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]

If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with: \[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \] \[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]

From the identified dynamics, compute \(\alpha\) and \(\beta\)

alpha = abs(dcgain(G('y', 'Fa')));
beta  = abs(dcgain(G('y', 'Fd')));

\(k_a\) is estimated using the following formula:

ka = 0.8/abs(dcgain(G('y', 'Fa')));

The factor can be adjusted to better match the curves.

Then \(k_e\) and \(k_1\) are computed.

ke = ka/(beta/alpha - 1);
k1 = 1/beta - ke*ka/(ke + ka);

The damping in the system is adjusted to match the FEM model if necessary.

c1 = 1e2;

Analytical model of the simpler system:

Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
     [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
      -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];

Ga.InputName = {'Fd', 'w', 'Fa'};
Ga.OutputName = {'y', 'Fs'};

Adjust the DC gain for the force sensor:

F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));

We now compare the FEM model with the simplified simscape model.

We save the parameters of the simplified model for the APA300ML:

save('./mat/APA300ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');

We first extract the stiffness and mass matrices.

K = extractMatrix('mat_K_6modes_2MDoF.matrix');
M = extractMatrix('mat_M_6modes_2MDoF.matrix');

Then, we extract the coordinates of the interface nodes.

[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');

save('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

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

The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.

The flexor is now imported into Simscape and its parameters are estimated using an identification.

The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor. And we find the same parameters as the one estimated from the Stiffness matrix.

Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 18.

The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.

Kx = K(1,1); % [N/m]
Ky = K(2,2); % [N/m]
Kz = K(3,3); % [N/m]
Krx = K(4,4); % [Nm/rad]
Kry = K(5,5); % [Nm/rad]
Krz =  K(6,6); % [Nm/rad]

The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model. The two obtained dynamics are compared in Figure

We first extract the stiffness and mass matrices.

K = readmatrix('mat_K.CSV');
M = readmatrix('mat_M.CSV');

Then, we extract the coordinates of the interface nodes.

[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');

save('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

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

The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.

The flexor is now imported into Simscape and its parameters are estimated using an identification.

The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor. And we find the same parameters as the one estimated from the Stiffness matrix.

Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 18.

The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.

Kx = K(1,1); % [N/m]
Ky = K(2,2); % [N/m]
Kz = K(3,3); % [N/m]
Krx = K(4,4); % [Nm/rad]
Kry = K(5,5); % [Nm/rad]
Krz =  K(6,6); % [Nm/rad]

The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model. The two obtained dynamics are compared in Figure

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');

The transfer function from the force actuator to the force sensor is identified and shown in Figure 23.

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);

The controller is defined and the loop gain is shown in Figure 24.

Kiff = -1e12/s;

Flexible joints have 0.25mm width.

We first extract the stiffness and mass matrices.

K = readmatrix('mat_K.CSV');
M = readmatrix('mat_M.CSV');

Then, we extract the coordinates of the interface nodes.

[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');

save('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

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

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]

na = 2; % Number of stacks used as actuator
ns = 1; % Number of stacks used as force sensor

The dynamics is identified from the applied force to the measured relative displacement. The same dynamics is identified for a payload mass of 10Kg.

m = 10;