Just remember that the first time history plot contains only rout (the output of the attitude controller to the mixer) and the sweep signal that is included in rout. I did not include the rate response of the vehicle. So here is a time history of the Y6 frequency sweep in the roll axis (shown below).

The red is the output from the attitude controller to the mixer (rout) and the green is the roll rate response of the vehicle. The scale for Rout is on the left and the scale for the roll rate is on the right. I think it is pretty clear that the signal to noise is sufficient to support what is seen in the bode plot for 1 rad/s to 5 rad/s. Not only is the SNR good, it also shows that the aircraft is 180 deg out of phase with the input for the low freqency regime. I will go out and gather another sweep in the low frequency regime but I think the analysis is consistent with what is observed in the time history plot.

well, what I only wanted to mention is that in all plots there is a loss of coherence below ca 0.8 Hz or so ā¦ and I think itās obvious why from looking at the data in the first few cycles (the last plot just reinforces it) ā¦ thatās all I wanted to mention

No. Iām glad you wrote. It made me look at the trace and convince myself that the results from 1 to 5 rad/s are consistent with the time history. just a clarification. You mention 0.8 Hz. Did you mean Hz or Rad/s?

I guess I wouldnāt expect good coherence below 1 rad/s since I didnāt run my sweep below 1 rad/s. Or are you speaking of frequencies above 1 rad/s where the coherence is good near 1?

ah ā¦ I missed that, sorry (I was assuing that the lowest shown frequency was the starting frequency)
so the plots below 1 rad/s are indeed not real, but for a trivial reason
I guess you want then to cut out the unreal parts in the plots

I am very confused about that low frequency phase.

Edit: I know what it is!!! It is propeller flapping torque. The aircraft is speeding up as it is tilted over and the propeller flapping torque is then being countered by the output of the rate loop.

In short for low frequencies you are characterising the relationship between propeller flapping torque or rate_out and airspeed. At higher frequencies airspeed can not build up and you are looking at the relationship between rate_out and rate.

Here is an update on the work that Iāve been doing. Just got back from a course on systems identification using a program called CIFER (Comprehensive Identification from Frequency Responses). I posted previously the pitch and roll frequency responses. I had also done frequency sweeps in the yaw axis but had not posted the results. I had the data with me and wanted to put what I had just learned to use. This course walked us through what was important about gathering the data (instrumentation, data consistency, frequency sweepsā¦), developing the bode plots that show the relationships between the important control inputs and outputs (states), and lastly how to develop a 6 degree of freedom model from the frequency response data. One thing that I learned/didnāt realize is how uncoupled multicopters are compared to helicopters. Helicopters become even more difficult to model going from a teetering rotor head to something that had geometric or effective offset (i.e. flap hinges displaced from the center of rotation or no flap hinges). So coming up with a model (set of transfer functions) that represent a multicopter is not too difficult. What remains to be seen is how scalable they would be to different multicopter size and configurations.
I used my frequency sweeps for the Y6 in roll and yaw to come up with a bare airframe model for the lateral/directional equations of motion. So you have 3 equations of motion for rigid body motion instead of the 6. SO the lateral directional equations are the lateral force, roll moment, and yaw moment equations.
I used CIFER to fit the following frequency responses, roll rate Ā§ to Rout (roll mixer output), lateral acceleration (AY) to Rout (roll mixer output), body axis lateral translation velocity (V) to Rout (roll mixer output) and yaw rate Ā® to Yout (yaw mixer output). It also determined the modes of motion for the system and stability and control derivatives. Here are the plots of the frequency responses with the model laid over it.
first chart is V to Rout and P to Rout

Generally the model is fitted for frequencies from 1.5 rad/s to 15-20 rad/s. It captures the unstable long term mode, the yaw short term mode, and the roll mode. There was no roll damping derivative which leads me to believe the roll mode is effectively acceleration command in the short term. Here are the transfer functions

So in order to develop a full 6 DOF model, I will need to do a sweep in the vertical axis with the collective (thatās throttle for the multiās). Then I can use the same process. So this information could be used to develop better a better control system however these equations only apply to the Y6. So a lot more work would need to be done to understand how to build a generic model of a multicopter or helicopter and what physical attributes for that type of aircraft play into the eigenvalues or the stability and control derivatives.

Reopening this thread with some new data. @ChrisOlson and I were able to get some frequency response data on two fairly different helicopters. I think things are starting to make sense with regard to the RC helicopter frequency response. So here are the vehicles
Billās
Ergo .46 converted to electric
E-flite 700 heli motor
Castle Creation 80Amp 50VDC ESC
12s 5000mAh Lipo batteries
rotorspeed: 1800 RPM
blades: 600mm
rotorhead: Miniature Aircraft X-cell 30 2 bladed flybarless (I removed the flybar)
Dampers in rotor head are pretty soft.
weight: 10.5 lbs
swashplate servos: Futaba 3050 digital servos

Chrisā Heli
Synergy E5 stretch
Iām assuming it is running 626mm blades
Flybarless head with stiff dampers
low head speed 1550 rpm
high head speed 1700 rpm
Iāll let Chris fill in the rest. Iām sure his is a little lighter than mine.

Here is the comparison of the two aircraft in pitch. This plot is a bode plot of pitch rate to pout which is the pitch command being sent to the mixer. I plotted coherence there for completeness but tried to trim the plot on the x axis to exclude regions where the coherence was poor.
ergo is labeled ergofbl
626 low headspeed is labeled olson03
626 high headspeed is labeled olson04

this shows that the bare airframe response of these RC helicopters is nearly rate command because the phase at lower frequencies is near 0 deg. For the Ergo this would be more difficult to fly because of the phase lag but for the 626 an experienced heli pilot could fly this aircraft.

I believe the rotor head dampers play a big part in how far out the mag and phase go before they drop off. Rotor speed has some effect but for the RC heliās with teetering heads, however the damper plays bigger part.

The rigid body pitch mode of the aircraft can be determined and almost perfectly fits a second order system model for the frequencies shown here. The transfer function of the form

Yields the following data for the bode plot above

I will continue with the roll results at a later time.

This is 25 rad/sec not hz. So about 4 hz. I have believe we are seeing more lightly damped rigid body modes than the full scale Helis. Full scale helicopters with teetering rotors like the jet ranger (Bell 206) or the older H-1ās (also Bell product) can be modeled as a first order system for the roll and pitch rigid body modes. Even some of the lower geometric offset designs like the H-3. Geometric offset is the distance from the shaft to the flapping hinge. This helps increase the maneuverability of the aircraft. As the rotor system gets stiffer like in hingeless designs (i.e. Virtual flapping hinges) like the BO-105, the rotor response starts to couple with the body of the aircraft and the rigid body modes become second or higher order. Most full size helicopters have higher inertia rotor systems and thus the rigid body modes are fairly well damped (damping mainly coming from the rotor system).
So with model helicopters, the desire is highly maneuverable aircraft with good rotor response. So the rotor blades for the flybarless designs are lighter with less inertia and even though they are teetering rotor systems they have stiff dampeners that help drive up the natural freq. This means the rigid body modes are less damped and have higher natural frequencies.
So Iāve been reading Tischlerās book on helicopter system identification. Using the transfer function results, some rotor system characteristics can be determined. The rotor flapping time constant, natural frequency and control system delay. For Chrisā 626, it is 0.084 sec and for my ergo, it is 0.052 sec. seems odd that mine is quicker than chrisā which has the stiffer dampeners but they really impact the natural freq more than anything which is very obvious in this case. The Delay column in my previous post is the control system delay. You can see that my delay is much higher. Iām using digital servos but they are probably not as powerful as they should be.
So now the challenge is designing a control system that can better handle a lightly damped system. I think the current one does pretty well but maybe we can do better.
Regards,
Bill

Another update on the comparison of rotorheads for flybarless helicopters. So I bought a Synergy 626 which is the same aircraft Chris Olson had provided data for my comparison for the pitch axis. However the frequency sweep was cut off a little early and did not contain enough data to determine the natural frequency for the roll axis of the synergy aircraft. So I did the frequency sweep testing with my aircraft and the pitch axis looks nearly identical to Chrisā results. So here is a comparison of the Ergo and Synergy aircraft in the roll axis. The bode plot is frequency response of the roll rate to Rout signal.

So I did not get the two rotor speeds that Chris had gotten but I think this shows the difference in rotor heads. I think rotor speed plays a minor role and head stiffness is more important when it comes to the rotor flap natural frequency. In a hingeless rotor system, I believe rotor speed will have a more pronounced effect. I have a 4 bladed head and intend to show results from that when I get a chance to swap it out on one of these helicopters.
Here is the data based on the 2nd order model described in my previous post.

Notice that both have a lightly damped rigid body mode however the synergy natural frequency is double that of the Ergo.

Another update to this thread. Over the past month, I have been conducting tests looking at different rotor heads on the same helicopter. I used the Synergy E5 stretched to a 626. The same as I described above. I conducted frequency sweeps on the stock teetering rotor head for both 1500 and 1900 RPM. I removed the stock rotor head and replaced it with a 4 bladed rotor head from century helicopters. This is a hingeless rotor head in the flap degree of freedom and pinned in lead-lag like most model helicopters.

Here is a plot of both the pitch and roll frequency response for 1500 and 1900 RPM. Here is the description of the legend label
syn4b15p - 4 bladed rotor, 1500 RPM, Pitch axis
syn4b19p - 4 bladed rotor, 1900 RPM, Pitch axis
syn4b15r - 4 bladed rotor, 1500 RPM, Roll axis
syn4b19r - 4 bladed rotor, 1900 RPM, Roll axis

The data from the 4 bladed head shows the rotor regressive flap mode in both the pitch and roll axes. In the pitch axis the magnitude and phase roll off because the lead lag regressive is not coupled to the rigid body pitch mode. In the roll axis, the lead lag regressive is coupled with the rigid body roll mode and the magnitude and phase roll off after the lead lag mode at ~80 rad/s. The flap regressive is not as evident in the teetering head shown in my previous posts.
So my thought is that to suppress these lightly damped modes, notch filters will be needed for both. The more rigid the head the better. It keeps the flap regressive mode above the piloting frequencies (< 3 Hz ). This is now more evident with my X-3 which uses the 4 bladed rotor system. The flap regressive coupled with the body was around 5 hz which allowed me to notch that frequency and enabled me to get he P gain up to 0.2 which is unheard of for heliās. The aircraft handled much differently than heliās that are tuned with the wiki method. So the goal is to get the flap regressive freq higher which may require stiffer dampeners in the rotor head.