**Procedures
for Unimorph Actuator Characterization:**

Here, the actuator is modeled as a mass-spring-damper system as shown below.

(NB: When steady-state measurements are made, only stiffness parameters appear (dampers in the model can be removed and masses simply become rigid links).

1) Ensure that there is no short circuit across the electrodes. Often, the short can be fixed simply by running a knife blade over the edges where they may occur.

2) Pole the actuator with positive polarity on the piezo side. Poling voltage will depend on piezo material and thickness (e.g., typically pole 0.125mm thick PZT at 300V).

3)
Ensure actuator base is rigidly mounted at base before taking
measurements. Apply 150V, then measure and
record the displacement, x_{DC} of the actuator distal end. NB: Using the highest magnification on the
appropriate lens and observing the motion on the Sony Trinitron monitor, take
calibration as 1 mm on monitor=7.23 mm
actual.

4)
With piezo side facing upwards, note position of actuator
distal end for 0V. Apply a known mass,
m, to tip of actuator and measure the displacement, Dx. Calculate and record
stiffness as k_{a}=m g/Dx.

5)
Slowly increase the voltage to V_{a}, the point at
which the distal end returns to the original position. Calculate and record blocked force as F_{b}=m
g (150V/V_{a}). Remove the
mass.

6) Perform a frequency sweep to determine the resonant frequency keeping in mind the following:

a. To prevent overstraining, start with low voltage amplitude since Q may be high.

b. Apply a positive offset bias to prevent depoling of unimorph.

7)
Determine where V_{AC}
is the pk-pk voltage and x_{AC} is the displacement range.

8) The energy product @150V, effective actuator inertia and effective actuator damping will automatically be calculated:

**Procedures
for Thorax Characterization:**

Here, the dynamics for the actuator coupled to a single 4-bar is modeled as shown below.

1)
Apply 150V DC to both actuators and record the changes in
actuator displacements (x’_{DC}) and output spar angles (q’_{DC}). (NB: The effect of differential coupling between the two spars is
reduced by driving both actuators simultaneously.) The transmission ratio (T) and structure parallel stiffness (both
in actuator coordinates and output spar coordinates) are automatically
calculated: . Return the voltage
to zero.

2)
To determine the serial compliance, move the wing by a known angle
(q’’_{DC}) and measure the
resulting change in actuator displacements (x’’_{DC}). (NB: Move wing in same direction as the
actuator would drive it.) The series
spring force can be calculated as F_{s}=(k_{a}+k_{p})x’’_{DC}. If there were no serial compliance, the
actuator should really have been displaced a distance q’’_{DC}/T so the series spring constant can be
calculated as follows: . Remove the external
load on the wing.

3)
To determine the differential parameters, apply a known
differential voltage (DV) to the two
actuators. Measure the resulting
actuator displacements (x’’’_{DC}) and the wing rotation angle (f’’’_{DC}). Determine the actuator displacement discrepancies as compared to
the case with no rotation, the corresponding series spring force and
displacement (e.g., if V_{1} is
applied to the leading actuator, the discrepancy is ; the spring force is F_{s1}=(k_{a1}+k_{p1})Dx_{1} and the overall series spring
displacement is x_{s1}=Dx_{1}+F_{s1}/k_{s1}). Equating the energy stored in the wing
rotation with the sum of these energies:
.

**CAVEAT:** This energy method really only works if the series
spring forces are zero when the rotation angle is zero (equivalently, if the
differential was removed, the equilibrium spar positions would remain
unchanged). Otherwise, the change in
energy must take into consideration the pre-stressed state).

4) Drive the actuators in flapping mode (i.e., in phase) and sweep the frequency to determine the resonant frequency (observe the same precautions as in step (6) for the actuator characterization). Measure the output angle amplitudes for each spar.