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University of GlasgowDEPARTMENT OF
AEROSPACEENGINEERING
Analysis of Factors Influencing Gyroplane
Longitudinal Stick Position and Gradient
Dr. Stewart Houston
Dr. Douglas Thomson
Dept, of Aerospace Engineering University of Glasgow
Analysis of Factors Influencing Gyroplane
Longitudinal Stick Position and Gradient
Dr. Stewart Houston
Dr. Douglas Thomson
Departmental Report No. 0005
December 2000
Report No. 0005 Houston/Thomson
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Dept, of Aerospace Engineering University of Glasgow
Introduction
Stick position and stick gradient as functions of airspeed, can be important indicators
of aircraft handling qualities. Gradient is a measure of the stability derivative Mu, which has
an important role to play in determining the period of phugoid-type oscillations. Since BCAR
Section T dynamic stability criteria are predicated on oscillation characteristics, stick position
and gradient may be a significant indicator or check of compliance. Stick position assumes
increased importance for the gyroplane because it is one of the few parameters that can be
readily measured without sophisticated or dedicated instrumentation.
Report No. 0005 Houston/Thomson
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BackgroundSimulation and flight test of the VPM Ml 6 have shown the importance of vertical c.g.
position in determining the stability of the phugoid-type oscillation of gyroplanes. Raising
the c.g. tends to confer positive angle-of-attack stability {Mw <0) which tends to stabilise
the phugoid oscillation, which is likely to mean compliance with Section T. However, during
technical audit of the work, it was pointed out by Colin Massey, GKN-Westland’s Chief
Aircraft Performance Engineer, that raising the c.g. could have an adverse impact on stick
gradient, and that this could set a limit on the extent to which the c.g. could be raised. Colin
Massey used a graphical approach to demonstrate the case, see Appendix. This Report however outlines an alternative approach.
Report No. 0005 Houston/Thomson
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Dept, of Aerospace Engineering University of Glasgow
Analysis
The analysis is based on an alternative mathematical method. This provides a
dissimilar verification of the graphical approach contained in the Appendix.
The longitudinal aircraft equations of motion for equilibrium flight, incorporating the
same schematic as the graphical analysis, and referring to the Figure 1 below, are given by
Figure 1 - Schematic of longitudinal forces and moments
= Tp-Tsmr]- mgfmO = 0
2Z = -T COST] + mg cos 6 = 0
= Tphp + Ts'mrihr - TcosT]jcr = 0
Report No. 0005 Homton/Thomson
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These equations are now used to examine stick position with airspeed. The
independent variable is not pitch attitude, but rotor angle of attack. (Glauert's original work
remember showed rotor thrust to be a function of axial velocity - a constant axial velocity at
constant rotorspeed can only be achieved with increasing airspeed if the angle of attack is
reduced). We have therefore sought to express our governing equation in terms of this
parameter. It can be shown easily from the above equations of motion, assuming small angles
and neglecting rotor flapping, that
7] =K
where 77 is the tilt of the rotor thrust vector (positive aft), xr is the position of the c.g. ahead
of the rotor hub in body axes, hp is the height of the rotor hub above the c.g. in body axes, hr
is the height of the c.g. above the propeller thrust line in body axes and ar is the rotor disc
angle of attack (we further assume that rotor thrust is normal to the disc). From simple
geometric considerations
ar = 7] + e (2)
We can now engage in some analysis.
From equation (1), it can be seen that if the propeller thrust line passes through the
c.g., tilt of the rotor thrust vector (i.e. stick position) is independent of rotor angle of attack
and therefore airspeed.
From equation (1), for a given ar i.e. a given airspeed, a configuration with the
propeller thrust line below the c.g. {hp>Q) will have a smaller value of 77 than for a
configuration with the propeller thrust line above the c.g. {hp < 0), i.e. the stick will be
further forward. This is consistent with RASCAL model simulation results, as can be seen in
Figures 12 & 18 of the Air Command Report.
Report No. 0005 Houston/Thomson
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From equation (1), for ar reducing (i.e. increasing airspeed), then 77 increases i.e. the
stick moves aft if the c.g. is above the propeller thrust line {hp> 0). Conversely 77 decreases
i.e. the stick moves forward if the c.g. is below the propeller thrust line {hp> 0).
It can therefore be concluded that our mathematical analysis is consistent with the
graphical analysis in the Appendix.
The veracity of this qualitative assessment can now be challenged quantitatively,
without recourse to more diagrams which is the only alternative if no equation is available.
(a) From the flight test data available for the VPM Ml6, at 24 knots (lowest
airspeed achieved), ri=62.9%=l 1.2deg, and 0=5.3deg. From equation (2), this gives ar
=16.5deg. Using equation (1) then gives hp =-0.23m.
(b) From the flight test data available for the VPM Ml 6, at 40 knots
ri=55%=9.8deg, and 0=3.4deg. From equation (2), this gives ar =13.2deg. Using
equation (1) then gives hp =-0.13m.
We have limited ourselves to 40 knots as above this speed, the assumptions in the model are
probably no longer valid (this is true in the graphical approach as well). Both of these values
of hp indicate a c.g. well below the propeller hub axis, substantially more than has been
measured (0.03m below at light weight, 0.06 below at high weight). Accordingly, this
mismatch indicates the limiting nature of the underlying assumptions (even at low speed
where they are most valid) made in the development of the simple model in equation (1),
assumptions common with the Appendix.
(c) A further interesting insight can be obtained if we use the flight test data
above, to calculate stick position for two nominal values of hp, of ±2 inches.
Report No. 0005 Houston/Thomson
Dept, of Aerospace Engineering University of Glasgow
A/s (knots) c.g. 2in above prop, thrust c.g. 2in below prop, thrust
24 T|=8.04deg Ti=9.18deg
40 T|=8.16deg Ti=9.07deg
So it is clear that indeed, the stick gradients are as the graphical method predicts, further
verifying that assessment. However, in quantitative terms, a ±2 inch variation in vertical c.g.
position has a very small impact on stick gradient, about 0.1 deg over 16 knots, which is
about 0.6% of stick travel, or about 0.005 in/knot, i.e. one or two orders of magnitude smaller
than any of the configurations shown on Figure 2, although it has a more significant impact on
stick position (at either airspeed above, of the order of 1 deg or about 6% of stick travel).
These figures double if one assumes ±4 inch variation in vertical c.g. position, which is really
quite extreme. So, although the assessment regarding stick gradient is correct, it would appear
that the gradient is relatively insensitive to vertical c.g. position, although stick position is
not.
-RAF 2000 - - -O- - VPM M16 — ■ -Air Command
10 30 50 70 90
-0.02--0.04- jO-----□--0.06-
i -0.08--0.1-
-0.12--0.14--0.16-
airspeed (knots
Figure 2 - Longitudinal stick gradient comparisons
The flight test data reproduced in Figure 2 is inconclusive since there appears to be no
correlation between stick gradient and vertical c.g. (as the quantitative analysis above would
tend to suggest). The RAF 2000 has a very low c.g. when dual as tested (about 9 in below the
propeller thrust line; the VPM has a thrust line close to or slightly below the c.g.; the
MODAC 503 has thrust line and c.g. almost coincident (it does tend to agree with the simple
Report No. 0005 Houston/Thomson
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analysis in this regard, as it has almost no stick gradient with speed relative to the others);
and by consent Roger Savage's Air Command measurements are at variance with Chris
Chadwick's.
Discussion
Given a desire to have something simple and easily understood by the wider
community, we would argue that our analysis results in ease of interpretation and clarity of
understanding that might be absent with the graphical interpretation (both in terms of the
number of figures, and their detail).
However, given that both approaches lead to the same conclusion, what does this
mean for any advisory Section T material? Moving the c.g. up relative to the propeller thrust
line, will tend to make Mw < 0, tending to stabilise the unstable phugoid oscillation. Leaving
aside the quantitative analysis above, the consequence on stick gradient is to flatten and then
reverse stick gradient, and tend to render Mu < Owith possible attendant consequences for the
phugoid. A compromise therefore seems desirable, but the quantitiative analysis above tends
to indicate that stick gradient and hence Mu is much less sensitive to vertical c.g. than is Mw.
However, it would be prudent to check as part of any design modification package.
Report No. 0005 Houston/Thomson
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Conclusion
Graphical and analytical approaches concur completely, speeifically that, subjeet to
approximation and assumption, the stick position gradient with speed is negative (i.e. stiek
forward with inereasing speed) for eonfigurations with the c.g. below the propeller thrust line;
and positive (i.e. stick aft with increasing speed) for configurations with the c.g. above the
propeller thrust line. However, it would appear that the gradient is relatively insensitive to
vertical c.g. position, although stick position is not.
Report No. 0005 Houston/Thomson
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Appendix
Report No. 0005 A-1 Houston/Thomson
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