MFI evolution

01η

Date of Fabrication
July 2001

Close up of the wing differential on 01η

Description

01η was constructed in August 2001 (the fourth attempt at building a 2 DOF differential using PZT actuators). The third attempt (which succeeded 00κ) was made to have a resonance of about 70 Hz, but turned out to have a resonance of only 35-40 Hz. This was not explainable by the previous dynamic model which assumed two seperate axes of flapping and rotation. A dynamic modeling of the differential based on a Lagrangian energy technique was undertaken which showed that the resonance depended on more than just the flapping and rotational inertia of the wing and identified an important cross-coupling inertia component of the wing. One important design change made at this time was that what was the "leading" spar in the original designs (the spar with the spherical joint at the end) now became the "lagging" spar. (See the figure above.)

Another important advance in this generation was to use polyimide wings. This method enables the fabrication of very light, stiff wings with very accurate inertias which was found to be very critical for getting the right response. For the first time, we had a wing which has the same inertia as a calliphora wing.

Salient Features

The structure attained the resonant frequency it was designed for to a high degree of accuracy.
The material used for flexures remained 6.25 um polyester. It was known that this will lead to a high differential stiffness which it turned out to have.
The nominal frequency response of the structure was validated to a high degree of confidence by multiple measurements both with strain gage measurements as well as with capacitive sensing experiments.
Inspite of the high coupling, τ showed a large rotation at high frequency (@100 Hz). The following movie shows the motion of the wing as seen under a strobe light when the structure was driven with 2 asymmetric triangular waves:

Note
1. The rotation is in the correct direction (i.e the wing has a positive angle of attack during its motion, or in other words, the leading spar acutally leads.)
2. The rotation at the ends is quite high. We can easily see a ±45 degree rotation if not more.
3. The rotation is of the type "delayed at the top" and "timed at the bottom".

Fabrication Parameters

Actuator Parameters
Two matched 16 x 3 PZT-5H actuators were used.

Blocking Force: 85 mN @ 200V
DC deflection (unloaded): 750 microns @ 200V
Actuator inertia reflected onto wing hinge: 3.3 mg-mm^2 (each)
Actuator stiffness reflected onto wing hinge: 0.95e-6 Nm/rad (each)

Fourbar Parameters

slider crank attachment point : 2 mm.
fourbar dimensions : 6, 6, 6, 1 mm.
Fourbar inertia reflected onto wing hinge: 1.7 mg-mm^2 (each)
fourbar parallel stiffness: (not measured, theoretically < 1/10 actuator stiffness)

Differential Parameters

6.25 micron polyester flexures (for all joints including in slider crank).
differential transmission ratio: λ = 2
size of differential:
1. length of leading spar : 3 mm, inertia = 3 mg-mm^2
2. length of lagging spar : 2 mm, inertia = 2 mg-mm^2

Length of flexures: (l_ψ = l_θ3 = 175 microns. all other flexures 50 microns each.)
Mean differential Stiffness: 0.47e-6 Nm/rad (about 1/2 of actuator stiffness)
Inertia of "middle" portion of differnetial:
md1 = 4.4, md2 = 0.37, md12 = 1.02 (mg-mm^2)

Wing Parameters

Wing Inertias
Jxx = 21.1505, Jxz = 5.6836, Jzz = 2.2875 (mg-mm^2)
where 'x' is the flapping direction and 'z' is the rotation direction. This inertia distribution was acheived after offsetting the original wing by 0.5 mm in the rotational direction. The wing was made using the new polyimide recipe. Face sheet thickness = 7 microns, tube dia = 200 microns. weight = 0.48 mg. inertia = 21 mg-mm^2.
Self resonance: ~ 350 Hz

Predictions

Nominal frequency response from torques to spar angles:

First Resonance at 105.5532 Hz
Second Resonance at 160.455 Hz
Note that the coupling exhibited is much larger than for a larger damping. The prediction above is for small amplitudes where Q ~ 4.5

Nominal frequency response predicted for strain gage measurements (the frequency response from voltage inputs to strain gage signals):

First Resonance at 120.314 Hz
Second Resonance at 180.478 Hz

Measurements

Resonant Frequencies (from Mimmo's data)
First resonance at 105 Hz
Second resonance at 160 Hz

Resonant Frequencies (from Rob's measurements)
First resonance at 115 Hz
Second resonance at 180 Hz
(the difference between the observed resonances can be theoretically explained by accounting for the strain gage setup dynamics)
Q: about 2.5 for large amplitudes (120+ V) and about 4.5 for smaller amplitudes (18 V)
DC motion: ~40 degrees for 150V

Comparisions

Theoretical and Measured Frequency response of strain gage signal:

From Vinput_lead -> Vsense_lead

From Vinput_lead -> Vsense_trail

Theoretical and Experimental Frequency responses from Mimmo's measurements:

Note that only the 1-1 measurement was made.