m 2001iMt

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.

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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.

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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

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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

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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.

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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.

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Appendix

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