3.3 Full order model
Piezoelectric models are described using both mechanical qmech and electric potential DOF V. As detailed in sections section 3.1 and section 3.2, one obtains models of the form
for both piezoelectric solids and shells, where Zqq(s) is the dynamic (mechanical) stiffness expressed as a function of the Laplace variable s.
For piezoelectric shell elements, electric DOFs correspond to the difference of potential on the electrodes of one layer, while the corresponding load is the charge Q. In SDT, the electric DOFs for shells are unique for a single shell property and are thus giving an implicit definition of electrodes (see p_piezo Shell). Note that a common error is to fix all DOF when seeking to fix mechanical DOFs, calls of the form 'x==0 -DOF 1:6' avoid this error.
For volume elements, each volume node is associated with an electric potential DOF and one defines multiple point constraints to enforce equal potential on nodes linked by a single electrode and sets one of the electrodes to zero potential (see p_piezo ElectrodeMPC and section ?? for a tutorial on how to set these contraints). During assembly the constraints are eliminated and the resulting model has electrical DOFs that correspond to potential, or differences of potential (if the other electrode's potential is set to 0) and loads to charge.
Short circuit (SC), charge sensors configurations correspond to cases where the potential is forced to zero (the electrical circuit is shorted). In (10), this corresponds to a case where the potential (electrical DOF) is fixed and the charge corresponds to the resulting force associated with this boundary condition.
A voltage actuator corresponds to the same problem with V=VIn (built in SDT using fe_load ('DofSet') entries). The closed circuit charge is associated with the constraint on the enforced voltage and can be computed by extracting the second row of (10)
| Figure 3.2: Short circuit: voltage actuator, charge sensor |
p_piezo ElectrodeSensQ provides utilities to build the charge sensors, including sensor combinations.
SC is the only possibly boundary condition to impose in a FEM model where voltage is the unknown. The alternative is to leave the potential free which corresponds to not specifying any boundary condition.
Open circuit (OC), voltage sensor, configurations correspond to cases where the charge remains zero and a potential is created on the electrodes due to mechanical deformations.
A piezoelectric actuator driven using a charge source also would correspond to this configuration (but the usual is voltage driving).
The voltage DOFs {V} associated to open-circuits are left free in (10). Since electrostatics are normally considered, ZVV is actually frequency independent and the voltage DOFs could be condensed exactly
This configuration is to be used for a voltage sensor, for example when the piezoelectric transducer is attached directly to the data acquisition card or a voltage amplifier (with very large impedance for sensing). In both cases the impedance is very large leading to a configuration close to an open circuit (OC, infinite impedance). Another example of OC boundary conditions is the use of current (charge) amplifiers for actuation, which is rarely used in practice but possible.
| Figure 3.3: Open circuit (voltage sensor, charge actuator) |
Since voltage is an explicit DOF, it can be observed using fe_case('SensDof')
sensor entries. Similarly charge is dual to the voltage, so a charge input would be a simple point load on the active DOF associated to an electrode. Note that specifying a charge distribution does not make sense since you cannot both enforce the equipotential condition and specify a charge distribution that results from this constraint.
It is possible to observe charge in an OC condition, but this is of little interest since this charge will remain at 0.
In summary, when computing modes under voltage actuation, the proper boundary condition is a SC, while for current (charge) actuation, it would be OC. For sensing, a voltage sensor corresponds to OC, while a charge sensor requires SC.
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