Piezoelectric sensors and actuators

SDT-piezo Contents Functions

4.6 Piezoelectric sensors and actuators

4.6.1 General theory

Essentially, when using piezoelectric materials in finite element models of structures, additional electrical degrees of freedom are added to the model, generally in the form of voltage degrees of freedom. The general form of the equations of motion was already given (see (9)) and is recalled below:

(4)

The equations of motion couple the electrical DOFs to the mechanical ones through the matrices K_Vq and K_qV.
When piezoelectric materials are used for sensing and actuating, they have associated electrodes which enforce equipotentiality on the surfaces. In the finite element model, this is taken into account by adding equality constraints between all the electrical DOFs associated to an electrode. One single electrical DOF is then linked to each electrode, and these DOFs are used for actuation and sensing.
Note that in the shell formulation of SDT, the electrical DOFs are not associated to nodes, but rather to elements, as detailed in section 3.2. Each element can have several layers among which some can be defined with piezoelectric material. Each piezoelectric layer of the shell element has then one associated DOF which corresponds to the difference of potential between the top and the bottom electrode of that layer. This is equivalent to considering that the bottom electrode of each layer is set to 0 volts and that the electrical DOF represents the voltage on the top electrode.
For 3D elements, each node has an additional associated electrical DOF, so that it is necessary to define boundary conditions in terms of voltage on the two electrodes. In SDT, electrical DOFs are identified with .21.
As explained in section 3.3, actuation can be of two types: an electrical charge Q can be imposed on one of the electrodes, which is equivalent to defining a mechanical force, as this term is on the right-hand-side of the equations. This case should be treated as the addition of point loads using fe_case('DofLoad') calls. It is however more common to use piezoelectric actuators in a voltage-driven mode, which corresponds to imposing the voltage, and can be treated using a fe_case('DofSet').
Assume that the voltage DOFs can be divided in two parts :

(5)

where V_in corresponds to the voltages imposed on the actuators in the model, and the other voltage DOFs are left free. We assume that the constraints related to the electrodes have already been taken into account to reduce the different matrices. The first line of the equations of motion becomes (assuming K_VV is a diagonal matrix, which is usually the case as it corresponds to the matrix of the individual capacitances) :

(6)

The fact that there is no coupling term in the mass and damping matrices leads to an equivalent problem where an imposed voltage corresponds to a mechanical load −[K_{qV_in} ] {V_in}.
Piezoelectric sensors are also of two types. Voltage sensing corresponds to measuring the voltage DOF directly, and can thus be defined by a simple fe_case('SensDOF') call. If a charge amplifier is used to measure the signal generated by the piezoelectrc sensor, this is equivalent to measuring a reaction force at the associated electrical DOF. In order to do that, one first needs to fix the associated electrical DOF using a fe_case('FixDOF') (if set to 0) or a fe_case('DofSet') (if used both as voltage actuator and charge sensor as in the case of self-sensing and piezo-shunt applications) call, and then define the charge sensor with a p_piezo ElectrodeSensQ call. Assume again that the electrical DOFs are divided in two parts

(7)

where V_q are fixed electrical DOFs where the charge needs to be measured. The charge on these electrodes is then given by:

(8)

The definition of actuators and sensors of these different types are illustrated below for a plate modelled with shell elements, and an example of an accelerometer modelled with 3D elements.

4.6.2 Aluminum plate with 4 PZT patches (Shell model)

The first example deals with an aluminum plate with 4 piezoceramic patches. The geometry and the material properties are given in Figure 4.19 and Table 4.1

Figure 4.19: Geometric details of the aluminum plate with 4 piezoceramic patches

Property Value

Aluminum plate

E 72 GPa

ν 0.3

ρ 2700 kg/m³

Piezoceramic patches

E 54.05 GPa

ν 0.41

ρ 7740 kg/m³

thickness 0.25 mm

d₃₁ -185 10⁻¹² pC/N (or m/V)

d₃₂ -185 10⁻¹² pC/N (or m/V)

є₃₃^T 1850 є₀

є₀ 8.854 10⁻¹² Fm⁻¹

Table 4.1: Material properties of the plate and the piezoceramic patches

It corresponds to a cantilevered plate with 4 piezoelectric patches modeled using the p_piezo Shell formulation.
The first step d_piezo('TutoPzPlate4pzt-s1') consists in the creation of the model, the definition of the boundary conditions, and the definition of the default damping coefficient. The different meshing procedures are detailed further in section 5. The resulting mesh is shown in Figure 4.20

Figure 4.20: Mesh of the composite plate. The different colors represent the different groups

One can have access to the piezoelectric material properties (Figure 4.21) and the list of nodes associated to each pair of electrodes (Figure 4.22) using p_piezo Tab calls. Here nodes 1055 to 1058 are associated to the four pairs of electrodes defined in the model. The corresponding degree of freedom is the difference of potential between the electrodes in each pair corresponding to a specific piezoelectric layer. In this models, layers 1 and 3 are piezoelectric in groups 2 and 3 (the internal layer correspond to supporting aluminum plate). Therefore only .21 (electrical) DOF is associated to nodes 1055-1058.

p_piezo('TabDD',model);      % List piezo constitutive laws

Figure 4.21: Example subset of table with the full set of mechanical, dielectric and piezoelectric coefficients in the 4 different forms of the constitutive equations

r1=p_piezo('TabInfo',model); % List piezo related properties

Figure 4.22: Information about electrical master nodes related to each piezoelectric layer. Here layers 1 and 3 for two zones with ProID 104 and 109 define the 4 piezoelectric patches

The next step d_piezo('TutoPzPlate4pzt-s2') consists in the definition of the actuators and sensors in the model. Here, we consider one actuator on node 1055 (layer 3 of group 1), the four piezoelectric patches are used as charge sensors, and the tip displacement of the cantilever beam is measured at the right-upward corner of the beam (corresponding to node 1054 here). Note that in order for Q-S1, Q-S2 and Q-S3 to measure resultant charge, the corresponding electrical difference of potential needs to be set to zero. If this is not done, then the charge sensors will measure a charge close to zero (round-off errors) as there is no charge when the difference of potential across the electrodes is free. For Q-Act, the electrical DOF is already fixed due to the fact that the patch is used as a voltage actuator.

%% Step 2 - Define actuators and sensors and visualize
nd=feutil('find node x==463 & y==100',model);
model=fe_case(model,'SensDof','Tip',{[num2str(nd) ':z']}); % Displ sensor
i1=p_piezo('TabInfo',model);i1=i1.Electrodes(:,1);
model=fe_case(model,'DofSet','V-Act',struct('def',1,'DOF',i1(1)+.21, ...%Act
    'Elt',feutil('selelt proid 104',model))); % Elt defined for display
model=p_piezo(sprintf('ElectrodeSensQ  %i Q-Act',i1(1)),model); % Charge sensors
model=p_piezo(sprintf('ElectrodeSensQ  %i Q-S1',i1(2)),model);
model=p_piezo(sprintf('ElectrodeSensQ  %i Q-S2',i1(3)),model);
model=p_piezo(sprintf('ElectrodeSensQ  %i Q-S3',i1(4)),model);
% Fix ElectrodeSensQ dofs to measure resultant (charge)
model=fe_case(model,'FixDof','SC*S1-S3',i1(2:end)+.21);
cf=feplot(model); fecom('view3')
cf.mdl.name='Plate_4pzt'; d_piezo('SetStyle',cf); feplot(cf);

We can now visualize the voltage actuator and the tip displacement sensor. Note that to visualize the piezoelectric actuator, it is necessary to associated a group of elements when defining it (see fe_case('DofSet') command above).

fecom('showfialpha')
fecom('proviewon')

% Arrow length and thickness
sdth.urn('Tab(Cases,Tip){Proview,on,deflen,20}',cf)
sdth.urn('Tab(Cases,Tip){arProp,"linewidth,2"}',cf)
fecom('curtabCase',{'Tip';'V-Act'})

Figure 4.23 shows the visualization of the tip sensor and voltage actuator.

Figure 4.23: Visualization of the tip sensor and voltage actuator

We can also visualize the charge sensors (Figure 4.24). Note however that in the case of plates, it is not possible to distinguish on the visualization if the patch is on the top or the bottom of the plate, so that Q-S2 and Q-S3 would give the same visualisation.

fecom('curtabCase',{'Q-S1';'Q-S2'})
cf.mdl.name='Plate_4pzt_QS1-2';d_piezo('SetStyle',cf); feplot(cf);

Figure 4.24: Visualization of the 2 charge sensors Q-S1 and Q-S2

In order to check the effect of the actuator, we compute the static response using the full model and represent the deformed shape (Figure 4.25) in d_piezo('TutoPzPlate4pzt-s3') .

Figure 4.25: Deformed shape under voltage actuation on one of the bottom piezoelectric patches

We also compute the transfer function between the actuator and the four charge sensors, as well as the tip sensor using the full model.

Figure 4.26 shows the different transfer functions from the voltage actuator to the tip displacement sensor (left) and to all charge sensors (right). The figure shows that Q-SAct shows an alternance of poles and zeros, but the smallest distance between the poles and zeros. This configuration corresponds to transfer functions used for shunting applications. For QS1 to QS3, the pole-zero alternance is lost due to the fact that the sensor and the actuator are not strictly collocated. The pole-zero distance is also very different, although if we are looking at the structure with the beam theory, the three FRFs should be identical. This demonstrates clearly the need for shell models, and the impact of the location of the sensor on the pole-zero pattern, and pole-zero distances.

Figure 4.26: Open-loop transfer function between V-Act and tip displacement (left), the 4 charge sensors (right)

4.6.3 Piezoelectric shaker with an accelerometer mounted on top (3D model)

The second example deals with an accelerometer (sensor) mounted on a piezoelectric shaker (actuator). The piezoelectric shaker consists of two steel cylindrical parts with a piezoelectric disc inserted in between. The base of the shaker is fixed and the piezoelectric element is used as an actuator: imposing a voltage difference between the electrodes results in the motion of the top surface of the shaker to which the accelerometer is attached (Figure 4.27).

Figure 4.27: Piezoelectric accelerometer attached to a piezoelectric shaker for sensor calibration

The piezoelectric properties for the sensing element in the piezoelectric shaker are given in Table 4.3. The actuating element has the same properties as the sensing element and is poled through the thickness, as the actuator.

Part Material E (GPa) ρ (kg/m³) ν

Wear plate Al₂O₃ 400 3965 0.22

Sensing element Piezo 54 7740 0.44

Proof mass Steel 210 7800 0.3

Table 4.2: Mechanical properties of the wear plate, sensing element and proof mass

Property Value

d₃₁=d₃₂ -185 10⁻¹² pC/N (or m/V)

d₃₃ 440 10⁻¹² pC/N (or m/V)

d₁₅=d₂₄ 560 10⁻¹² pC/N (or m/V)

є₃₃^T=є₂₂^T=є₁₁^T 1850 є₀

є₀ 8.854 10⁻¹² Fm⁻¹

Table 4.3: Piezoelectric properties of the sensing element

The mesh is generated with d_piezo('TutoPzAccShaker-s1') and represented in Figure 4.28. In the meshing script, both a voltage and a charge sensor are defined for the piezoelectric disk in the accelerometer for the top electrode, and the the bottom electrode potential is set to zero.
In d_piezo('TutoPzAccShaker-s2') , a voltage actuator is then added to the shaker top electrode, while the bottom electrode potential is set to 0. The shaker is mechanically fixed at the bottom.

Figure 4.28: Mesh of the piezoelectric accelerometer attached to a piezoelectric shaker

%% Step 2 - Define actuators and sensors
  % -input "In" says it will be used as a voltage actuator
model=p_piezo('ElectrodeMPC Top Actuator -input "Vin-Shaker"',model,'z==-0.01');
  % -ground generates a v=0 FixDof case entry
model=p_piezo('ElectrodeMPC Bottom Actuator -ground',model,'z==-0.012');
% Voltage sensor will be used - remove charge sensor
model=fe_case(model,'remove','Q-Top sensor');

The different electrodes in the model can be visualized (Figure 4.29) and each actuator/sensor can be visualized separately (Figure 4.30)

Figure 4.29: Top and bottom electrodes for shaker voltage actuator and accelerometer voltage sensor

% Visualize Vin electrode
fecom curtabcases Vin-Shaker %

% Visualize VSens electrode
fecom curtabcases 'V-Top sensor'

Figure 4.30: Shaker input voltage(left) and accelerometer output voltage(right)

We can have an overview of the electrodes and associated electrical DOFs (Figure 4.31) :

r1=p_piezo('TabInfo',model); % List piezo related properties

Figure 4.31: Electrodes and associated nodes/DOFs

In d_piezo('TutoPzAccShaker-s3') , the FRF between the voltage applied on the piezoelectric shaker and accelerometer response (voltage mode) is computed and represented in Figure 4.32, together with the transfer function for the displacement of the base of the accelerometer.

% End of script

Figure 4.32: FRF between the voltage applied on the piezoelectric shaker and accelerometer response (voltage mode), and the displacement of the base of the accelerometer

Property	Value
Aluminum plate
E	72 GPa
ν	0.3
ρ	2700 kg/m³
Piezoceramic patches
E	54.05 GPa
ν	0.41
ρ	7740 kg/m³
thickness	0.25 mm
d₃₁	-185 10⁻¹² pC/N (or m/V)
d₃₂	-185 10⁻¹² pC/N (or m/V)
є₃₃^T	1850 є₀
є₀	8.854 10⁻¹² Fm⁻¹


Part	Material	E (GPa)	ρ (kg/m³)	ν

Wear plate	Al₂O₃	400	3965	0.22
Sensing element	Piezo	54	7740	0.44
Proof mass	Steel	210	7800	0.3

Property	Value
d₃₁=d₃₂	-185 10⁻¹² pC/N (or m/V)
d₃₃	440 10⁻¹² pC/N (or m/V)
d₁₅=d₂₄	560 10⁻¹² pC/N (or m/V)
є₃₃^T=є₂₂^T=є₁₁^T	1850 є₀
є₀	8.854 10⁻¹² Fm⁻¹