Article on rotor tuning

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There's a great article on rotor management and tuning at "Airgyro Aviation" on facebook, go to posting section on there site.
 
Copypasted:


​​​​​​Study of vibrations occurring on the rotor
Introduction

A semi rigid 2-bladed rotor suffers from several sources of vibration which can be categorized into two general groups: those which have a frequency identical to the rotor revolutions (1/rev) and those which have twice that frequency (2/rev).
Note that a tight cyclic to rotor head linkage is very helpful to eliminate loose shake in the stick that does not represent actual rotor head vibrations.

1 per rev rotor shake:
or synchronous at about 6 hz or 360 rpm.
This is due to two sources: Mass dynamic imbalance, and aerodynamic imbalance – unequal lift on each blade – often referred to as tracking mismatch.
Aerodynamic imbalance is normally adjusted by shims under the teeter block of the rotor hub bar.

NOTE: Perfectly tracked blade tips does not guarantee aerodynamic balance. Symmetry of blade airfoil shape at each station from root to tip and symmetrical flexibility of each blade are factors.

2 per rev shake:
at about 12 hz or 720 rpm.
Can be due to several sources – primarily, mis-matched teeter undersling height relative to coned rotor vertical CG.
The undersling, i.e., the distance from the teeter bolt to the center of the rotor hub.
Other sources that should be eliminated are side-to-side slop on the teeter bolt – allowing the rotor to “flop” from fore to aft each rev. Also, friction in the teeter bearing should be minimized. 2 per rev shake is caused by forward movement requiring teeter action.

1. Identify which shake you have. You may have both. Eliminate 2-per rev shake by flying in a vertical or slow airspeed decent. This eliminates most of the 2-per rev shake and leaves only 1-per rev shake.

2. Minimize 1-per rev shake by mass balance adjustments with chord sideways adjustments and span wise adjustments with tip weights.

Note: that adding weight to the tips – span wise balance - also can change the tracking of the blade tips because it can tip the cone of the rotor toward the heavier blade. So, if the aerodynamic center of the rotor is not concentric with the rotor CG, chord wise adjustments are only a compromise and 1 per shake may not be able to be well reduced. If tracking adjustments are available, it is possible to move the aerodynamic center of the rotor to be concentric with the mass CG, but this can be a tedious trial and error effort.

Just to mention, Magni rotors are held very well aerodynamically and mass symmetrical from the production processes. That means the mass and aerodynamic centers are well aligned. In this case adjusting span wise balance with tip weights fine tunes both the span mass dynamic balance and the tracking. So, span dynamic balance can be expected to be best when the tracking is best. There are no tracking adjustments on the Magni rotor, so blade tracking is accomplished by adjusting tip weights.
A dynamic prop balancer works well on such precise rotors to mass balance the rotor.

2 per rev shake: This shakes the stick twice as fast as the 1 per rev shake observed at slow airspeed or in a vertical decent. 2-per rev shake increases with airspeed because this requires more teeter action. As mentioned, slop in the teeter bolt and/or friction in the teeter bearings can create two per rev shake when moving forward. Minimize these issues first. Then teeter undersling mismatch to the rotor coned CG may be the remaining 2-per rev shake issue.

I find a fairly simple way to see which way the undersling needs to go is as follows in flight. (Don’t do this if you are flying a pitch unstable gyro – the gyro must return to trimmed airspeed upon a pulse pitch input to the cyclic.)
From normal straight and level cruise, pulse the cyclic slightly aft to initiate a slight nose-raising, increased G load. Immediately, let go of the stick and observe the stick vibration. Do the same with the cyclic in the forward direction to induce a slightly decreased G load transient.

If the stick shake is the same in both directions for a short period during the transient, your undersling is about right.

If the stick shake gets better in one direction, and worse in the other direction transient, the undersling is not matched to the rotor coned CG.
The reason this tells you something is that the rotor coning angle changes during these transients and the rotor CG moves vertically relative to the teeter pivot.
When the gyro is experiencing a lesser load on the rotor during a nose lowering transient, the coned rotor CG is lowering and may be either a better or worse match to the teeter undersling position.

When the gyro is experiencing a higher load on the rotor during the nose raising transient, the rotor cones a bit more and the CG raises relative to the teeter bolt. If a raising nose transient increases the stick shake momentarily, the rotor CG is moving further above the teeter bolt indicating the undersling should be higher. Visa versa is true if the 2-per rev shake gets better upon this positive G-load transient.

You can’t completely eliminate 2-per rev shake. The rotor coning angle will change during wind and maneuver transients, and cyclic input momentarily moves the rotor CG off center from the spindle axis.
But, if the amount of 2-per rev shake induced in the above test is the same in each transient direction, the undersling height is probably about right.

In summary, nothing beats a quality rotor that is symmetrical between blades both aerodynamically and in mass distribution. Nothing beats a good dynamic balancer for fine tuning mass balance.

2/rev These vibrations are usually due to five potential causes.
a) Rotating drag, or the difference in drag between when the rotor is at 3-9 o’clock and 12–6 o’clock. This is inherent in all 2 bladed Rotors.
b) In-plane resonance. The in-plane natural frequency of the rotor is about 6 hz which is the same as the rotor rpm and this creates a 2/rev vibration.
c) Inertia around the spanwise blade to blade C of G axis. Due to the fact that the blade tip plane is not perpendicular to the rotor bearing axis there is a 2/rev moment at the teeter bolt acting in the bearing shaft. With an optimum undersling the 2/rev vibration caused by this moment is minimised but there is always some. Incorrect undersling (either too large or too small) increases the moment of inertia around the spanwise axis and increases the resulting 2/rev vibration. The best solution is to optimise undersling.
d) High friction in teeter bearings. This is seldom a problem on Eurogyros because they usually use needle or roller teeter bearings (except AutoGyro). This could be a problem if these bearings were badly corroded and started to seize up. This problem seems to be more common with smaller gyros with teeter bushings rather than bearings.
e) Tail heavy rotor blades. A blade that is heavily under balanced chordwise will have a cyclic twisting motion as it rotates causing a 2/rev vibration.

Elaboration:
Krzysztof Wronowski
Manufaktura Lotnicza
 
Thank you, Xavier, for copying this article from "Airgyro Aviation"
My two cents. The author writes:
"With an optimum undersling the 2/rev vibration caused by this moment is minimised but there is always some. Incorrect undersling (either too large or too small) increases the moment of inertia around the spanwise axis and increases the resulting 2/rev vibration. The best solution is to optimise undersling"

I calculated the evolution of the torque on the plate, according to the underslung. It givesΓ[SUB]h[/SUB] = 8 Ω[SUP]2 [/SUP]M sin a[SUB]1 [/SUB](sin² a[SUB]0[/SUB]. R[SUP]2[/SUP]/3 – sin a[SUB]0[/SUB] . R.h + h²) sin (2Ω t)
With a[SUB]0 = [/SUB]coning, a[SUB]1[/SUB] = longitudinal flapping, M = masse of one blade (kg), Ω = rotation speed (rd/s), R = rotor radius (m) and h = undersling (m)

It shows well that the minimum of this vibratory torque is obtained when theteeter bolt is on the line which encloses the centers of gravity of each blade. This was generally accepted Sans titre10.png



The amplitude of the movement of the masses dm, parallel to teeter bolt, during a flapping angle α1 is worth: sin a[SUB]1[/SUB]. (r sin a[SUB]0[/SUB] -– h) sin(2Ω t)
Their speed v along this axis is : v =-sin a[SUB]1[/SUB] .2 Ω . (r sin a[SUB]0[/SUB] - –h) cos(2Ω t)
Their acceleration is : a = -sin a[SUB]1[/SUB]. 4Ω[SUP] 2[/SUP]. (r sin a[SUB]0[/SUB] -– h) sin(2Ω t)
Therefore it requires an elementary force: dF = -4dm Ω[SUP]2[/SUP] sin a[SUB]1[/SUB] (r. sin a[SUB]0[/SUB] - h). sin(Ω t). Hence, with dm= (M/R). dr : dF = -4sin(Ω t).(M/R) . Ω[SUP] 2[/SUP]. sin a[SUB]1.[/SUB] (r. sin a[SUB]0[/SUB] - h) . dr
So, for two blades, the sum of these dF between 0 and R is F= -8M.Ω[SUP]2[/SUP] sin a[SUB]1[/SUB]. (½R sin a[SUB]0[/SUB] - h) sin(2Ω t). Sans titre1.png



This force also acts around the pitch pivot to z meter below. and adds his torque on the control plate
So, the total torque vibration 2/rev would totally disappears when the undersling lower that "optimum"
Here, If z = 0.2 m, the required force for canceled the stick vibration is 68 mm, instead of "optimum" 82 mm Sans titre1.png
 

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I ended up correcting the many typo, sign, and coefficient errors, and I think the theory of the optimum needs to be corrected in this way. This seems to better reflect the observations of an smaller ideal undersling , and also the more sensitivity to the chosen height.
View attachment Vibrations selon undersling.xls
 

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